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geometric_algebra [2018/06/08 09:25] – [Modern] pbkgeometric_algebra [2023/12/30 00:23] (current) – [Videos] pbk
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 //Eine Sammlung nützlicher Ressourcen// //Eine Sammlung nützlicher Ressourcen//
  
-Introduced by Hermann Grassmann and greatly expanded by William Kingdon Clifford during the 19th century, Geometric Algebras provide a proper abstract framework for the treatment of geometrical vector operations that extend naturally to general dimensions. Their concise conceptualization yields elegant and peculiarly coherent constructs, in contrast with the intricacies of Vector Calculus. Because of the geometric origin of their structures and close relation to quadratic forms they turn useful in a wide range of applications in theoretical and applied sciences across several fields.+Introduced by Hermann Grassmann and greatly expanded by William Kingdon Clifford during the 19th century, Geometric Algebras provide a proper abstract framework for the treatment of geometrical vector operations that extend naturally to general dimensions. Their concise concepts yield elegant and peculiarly coherent constructs, in contrast with the intricacies of Vector Calculus. Because of the geometric origin of their structures and close relation to quadratic forms they turn useful in a wide range of applications in theoretical and applied sciences across several fields.
  
 ===== Sites ===== ===== Sites =====
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 ==== Research ==== ==== Research ====
  
-  * [[http://geocalc.clas.asu.edu|Geometric Calculus Research & Development]] - //David Hestenes//, Arizona State University+  * [[http://geocalc.clas.asu.edu|Geometric Calculus Research & Development]] - //David Hestenes//, Arizona State University
 This web site is dedicated to perfecting a universal mathematical language for science, extending its applications and promoting it throughout the scientific community. It advocates a universal scientific language grounded in an integrated Geometric and Inferential Calculus. This web site is dedicated to perfecting a universal mathematical language for science, extending its applications and promoting it throughout the scientific community. It advocates a universal scientific language grounded in an integrated Geometric and Inferential Calculus.
  
-  * [[http://www.science.uva.nl/research/ias/ga|Geometric algebra (based on Clifford algebra)]] - //Leo Dorst, Daniel Fontijne//, Intelligent Autonomous Systems, University of Amsterdam.+  * [[https://staff.science.uva.nl/l.dorst/clifford|Geometric algebra (based on Clifford algebra)]] - //Leo Dorst, Daniel Fontijne//, Intelligent Autonomous Systems, University of Amsterdam
 Geometric algebra is a very convenient representational and computational system for geometry. We firmly believe that it is going to be the way computer science deals with geometrical issues. It contains, in a fully integrated manner, linear algebra, vector calculus, differential geometry, complex numbers and quaternions as real geometric entities, and lots more. This powerful language is based in Clifford algebra. David Hestenes was the among first to realize its enormous importance for physics, where it is now finding inroads. The revolution for computer science is currently in the making, and we hope to contribute to it. Geometric algebra is a very convenient representational and computational system for geometry. We firmly believe that it is going to be the way computer science deals with geometrical issues. It contains, in a fully integrated manner, linear algebra, vector calculus, differential geometry, complex numbers and quaternions as real geometric entities, and lots more. This powerful language is based in Clifford algebra. David Hestenes was the among first to realize its enormous importance for physics, where it is now finding inroads. The revolution for computer science is currently in the making, and we hope to contribute to it.
  
-  * [[http://geometry.mrao.cam.ac.uk/home/latest-research|Geometric Algebra - Latest Research]] - //Chris Doran, Anthony Lasenby//, University of Cambridge.+  * [[http://geometry.mrao.cam.ac.uk/home/latest-research|Geometric Algebra - Latest Research]] - //Chris Doran, Anthony Lasenby//, University of Cambridge
 Conference proceedings, journal papers, lectures and reports. Conference proceedings, journal papers, lectures and reports.
  
-  * [[http://kmr.nada.kth.se/wiki/Main/Papers|The Knowledge Management Research Group]] - KTH Royal Institute of Technology+  * [[http://kmr.nada.kth.se/wiki/Main/Papers|The Knowledge Management Research Group]] - KTH Royal Institute of Technology
-Publications of the KMR Group.+
  
-  * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/Cliff.html|Cliffordians or Cliffhangers]] Study Group on Geometric Algebra.+  * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/Cliff.html|Cliffordians or Cliffhangers]] Study Group on Geometric Algebra 
 +  
 +  * [[https://www.jstor.org/action/doBasicSearch?Query=%28%28geometric+algebra%29+OR+%28clifford+algebra%29%29|Geometric Algebra OR Clifford Algebra]] at JSTOR  
 + 
 +  * [[https://scholar.google.com/scholar?q="geometric+algebra"+OR+"clifford+algebra"|Geometric Algebra OR Clifford Algebra]] at Google Scholar 
 + 
 +  * [[http://paperity.org/search/?q="geometric+algebra"+OR+"clifford+algebra"|Geometric Algebra OR Clifford Algebra]] at Paperity
  
-  * [[https://scholar.google.com/scholar?q="geometric+algebra"+OR+"clifford+algebra"|Geometric Algebra OR Clifford Algebra]] at Google Scholar.+  * [[https://www.researchgate.net/search?q=%22geometric%20algebra%22%20OR%20%22clifford%20algebra%22|Geometric Algebra OR Clifford Algebra]] at ResearchGate
  
-  * [[http://paperity.org/search/?q="geometric+algebra"+OR+"clifford+algebra"|Geometric Algebra OR Clifford Algebra]] at Paperity.+  * [[http://math.stackexchange.com/search?tab=relevance&pagesize=50&q=%22geometric%20algebra%22%20isaccepted%3ayes|Geometric Algebra]] and [[http://math.stackexchange.com/search?tab=relevance&pagesize=50&q=%22clifford%20algebra%22%20isaccepted%3ayes|Clifford Algebra]] at MathExchange
  
-  * [[https://www.researchgate.net/search?q=%22geometric%20algebra%22%20OR%20%22clifford%20algebra%22|Geometric Algebra OR Clifford Algebra]] at ResearchGate.+  * [[https://www.quora.com/topic/Geometric-Algebra|Geometric Algebra]] topic at Quora
  
-  * [[http://math.stackexchange.com/search?tab=relevance&pagesize=50&q=%22geometric%20algebra%22%20isaccepted%3ayes|Geometric Algebra]] and [[http://math.stackexchange.com/search?tab=relevance&pagesize=50&q=%22clifford%20algebra%22%20isaccepted%3ayes|Clifford Algebra]] at MathExchange.+  * [[https://observablehq.com/search?query=geometric%20algebra|Geometric Algebra]] at Observable
  
-  * [[https://www.quora.com/topic/Geometric-Algebra|Geometric Algebra]] topic at Quora.+  * [[https://duckduckgo.com/c/Geometric_algebra|Geometric Algebra]] topic at DuckDuckGo
  
-  * [[https://duckduckgo.com/c/Geometric_algebra|Geometric Algebra]] topic at DuckDuckGo.+  * [[https://github.com/topics/geometric-algebra|Geometric Algebra]] topic at GitHub
  
 ==== General ==== ==== General ====
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 This page collects material related to the book //Die Wissenschaft der extensive Grössen oder die Ausdehnungslehre Erster Teil, die lineale Ausdehnungslehre// (1844) by Hermann Grassmann, which introduced for the first time basic concepts of what today is known as linear algebra (including affine spaces as torsors over vector spaces) and introduced in addition an exterior product on vectors, forming what today is known as exterior or Grassmann algebra. This page collects material related to the book //Die Wissenschaft der extensive Grössen oder die Ausdehnungslehre Erster Teil, die lineale Ausdehnungslehre// (1844) by Hermann Grassmann, which introduced for the first time basic concepts of what today is known as linear algebra (including affine spaces as torsors over vector spaces) and introduced in addition an exterior product on vectors, forming what today is known as exterior or Grassmann algebra.
  
-  * [[http://wiki.c2.com/?CliffordAlgebra|Clifford Algebra]] at the WikiWikiWeb.+  * [[https://www.jstor.org/stable/pdf/2369379.pdf|Applications of Grassmann's Extensive Algebra]] (1878) - //Professor Clifford//, American Journal of Mathematics, Vol. 1, No. 4, pp. 350-358 
 + 
 +>  I propose to communicate in a brief form some applications of Grassmann's theory which it seems unlikely that I shall find time to set forth at proper length, though I have waited long for it. Until recently I was unacquainted with the Ausdehnungslehre, and knew only so much of it as is contained in the author's geometrical papers in Crelle's Journal and in Hankel's Lectures on Complex Numbers. I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science. The present communication endeavors to determine the place of Quaternions and of what I have elsewhere called Biquaternions in the more extended system, thereby explaining the laws of those algebras in terms of simpler laws. It contains, next, a generalization of them, applicable to any number of dimensions; and a demonstration that the algebra thus obtained is always a compound of quaternion algebras which do not interfere with one another. 
 + 
 +  * [[http://wiki.c2.com/?CliffordAlgebra|Clifford Algebra]] at the WikiWikiWeb 
 + 
 +  * [[https://bivector.net|biVector.net]] - Geometric Algebra for CGI, Vision and Engineering 
 +Clifford's Geometric Algebra enables a unified, intuitive and fresh perspective on vector spaces, giving elements of arbitrary dimensionality a natural home
  
   * [[http://www.euclideanspace.com/maths/algebra/clifford|Clifford / Geometric Algebra]] - //Martin John Baker//   * [[http://www.euclideanspace.com/maths/algebra/clifford|Clifford / Geometric Algebra]] - //Martin John Baker//
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   * [[https://amathew.wordpress.com/2012/04/09/vector-fields-on-manifolds|Vector fields on manifolds]] - //Akhil Mathew//   * [[https://amathew.wordpress.com/2012/04/09/vector-fields-on-manifolds|Vector fields on manifolds]] - //Akhil Mathew//
-Associated to the Riemannian bundle TM there is a bundle of Clifford algebras, Cl(TM), such that the fiber at each x in M is the Clifford algebra Cl(T_x M).+ 
 +Associated to the Riemannian bundle TM there is a bundle of Clifford algebras, Cl(TM), such that the fiber at each x in M is the Clifford algebra Cl(T_x M).
  
   * [[http://www.science20.com/just_want_read_new_physics_especially_related_quantum_gravity/finding_alice_quaternion_looking_glass-79515|Finding Alice In The Quaternion Looking Glass]] - //Colin Keenan//   * [[http://www.science20.com/just_want_read_new_physics_especially_related_quantum_gravity/finding_alice_quaternion_looking_glass-79515|Finding Alice In The Quaternion Looking Glass]] - //Colin Keenan//
-(...) I will start in the Quaternion "looking glass" and find a way out to the Clifford Algebra CL(3,0) used by Joy Christian in his counter-example to Bell's Theorem.+ 
 +(...) I will start in the Quaternion "looking glass" and find a way out to the Clifford Algebra Cl(3,0) used by Joy Christian in his counter-example to Bell's Theorem.
  
   * [[http://dgd.service.tu-berlin.de/wordpress/vismathws12/2014/02/20/829|Introduction to projective geometric algebra]] - //Charles Gunn//   * [[http://dgd.service.tu-berlin.de/wordpress/vismathws12/2014/02/20/829|Introduction to projective geometric algebra]] - //Charles Gunn//
-At a student seminar at the TU Berlin in fall 2013, the following problem in plane geometry was posed, "Given a point lying on a line, and a second point lying on a second line, find the unique direct isometry mapping the first point/line pair to the second." 
  
-  * [[https://www.mathphysicsbook.com|Mathematics for Physics]] - //Adam Marsh// +> At a student seminar at the TU Berlin in fall 2013, the following problem in plane geometry was posed, "Given a point lying on a line, and a second point lying on a second line, find the unique direct isometry mapping the first point/line pair to the second." 
-This book is meant to complement traditional textbooks by covering the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics coursesThe idea is to provide an intuitive, visual overview of these mathematical toolswith guiding end goals including but not limited to spinors and gauge theories.+ 
 +  * [[https://www.physicsforums.com/insights/interview-mathematician-david-hestenes|Interview with a Physicist: David Hestenes]] (2016) - Physics Forums Insights 
 + 
 +  * [[https://probablydance.com/2017/08/05/intuitive-quaternions|Less Weird Quaternions]] (2017) - //Malte Skarupke// 
 + 
 +> I've always been frustrated by how mysterious quaternions are. (...) That is until recently, when I came across the paper "Imaginary Numbers are not Real – the Geometric Algebra of Spacetime" which arrives at quaternions using only 3D math, using no imaginary numbers, and in a form that generalizes to 2D, 3D, 4D or any other number of dimensions. 
 + 
 +  * [[http://marctenbosch.com/quaternions|Let's remove Quaternions from every 3D Engine]] (2018) - //Marc ten Bosch// 
 + 
 +> There is a way to represents rotations called a Rotor that subsumes both Complex Numbers (in 2D) and Quaternions (in 3D) and even generalizes to any number of dimensions. We can build Rotors almost entirely from scratchinstead of defining quaternions out of nowhere and trying to explain how they work retroactively.
  
-  * [[https://www.physicsforums.com/insights/interview-mathematician-david-hestenes|Interview with a Physicist: David Hestenes]] (2016) - Physics Forums Insights.+  * [[https://quaternionnews.commons.gc.cuny.edu|Quaternion Notices]] 
 +Quaternion Notices is dedicated to publishing news about quaternion conference sessions, seminars, and individual presentations; archives of related materials; links to quaternion resources.
  
 ==== Conferences ==== ==== Conferences ====
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   * [[https://www.ime.unicamp.br/~agacse2018|AGACSE 2018]] (2018) - The 7th Conference on Applied Geometric Algebras in Computer Science and Engineering, University of Campinas.   * [[https://www.ime.unicamp.br/~agacse2018|AGACSE 2018]] (2018) - The 7th Conference on Applied Geometric Algebras in Computer Science and Engineering, University of Campinas.
 +
 +  * [[https://s2019.siggraph.org/presentation/?id=gensub_345&sess=sess346|SIGGRAPH 2019]] (2019) - Los Angeles: [[https://bivector.net/PROJECTIVE_GEOMETRIC_ALGEBRA.pdf|Course notes for Geometric Algebra for Computer Graphics]] [[https://arxiv.org/pdf/2002.04509|arXiv version]] - //Charles Gunn//.
 +
 +  * [[http://www.cgs-network.org/cgi20|ENGAGE 2020]] (2020) - Empowering Novel Geometric Algebra for Graphics & Engineering Workshop, CGI 2020, Geneva (Switzerland).
 +
 +  * [[https://bivector.net/game2023.html|GAME2023 - Geometric Algebra Mini Event]] (2023) - DAE - Kortrijk, Belgium.
 ==== Book companion ==== ==== Book companion ====
  
-  * [[http://www.geometricalgebra.net|Geometric Algebra for Computer Science]] - //Leo Dorst, Daniel Fontijne, Stephen Mann//+  * [[http://www.geometricalgebra.net|Geometric Algebra for Computer Science]] - //Leo Dorst, Daniel Fontijne, Stephen Mann//
 Companion site to the book Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry (Morgan Kaufmann). Companion site to the book Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry (Morgan Kaufmann).
  
-  * [[https://sites.google.com/site/grassmannalgebra|The Grassmann Algebra Book]] - //John Browne//.+  * [[https://grassmannalgebra.com|The Grassmann Algebra Book]] - //John Browne//
 This is the companion site for the book "Grassmann Algebra: Exploring extended vector algebra with Mathematica" This is the companion site for the book "Grassmann Algebra: Exploring extended vector algebra with Mathematica"
 and for the Mathematica-based software package GrassmannAlgebra. and for the Mathematica-based software package GrassmannAlgebra.
  
-  * [[http://faculty.luther.edu/~macdonal/laga|Linear and Geometric Algebra]] and [[http://faculty.luther.edu/~macdonal/vagc|Vector and Geometric Calculus]] - //Alan Macdonald//, Luther College.+  * [[http://faculty.luther.edu/~macdonal/laga|Linear and Geometric Algebra]] and [[http://faculty.luther.edu/~macdonal/vagc|Vector and Geometric Calculus]] - //Alan Macdonald//, Luther College
 Geometric algebra and its extension to geometric calculus simplify, unify, and generalize vast areas of mathematics that involve geometric ideas. Geometric algebra is an extension of linear algebra. The treatment of many linear algebra topics is enhanced by geometric algebra, for example, determinants and orthogonal transformations. And geometric algebra does much more, as it incorporates the complex, quaternion, and exterior algebras, among others. Geometric algebra and calculus provide a unified mathematical language for many areas of physics (classical and quantum mechanics, electrodynamics, relativity), computer science (graphics, robotics, computer vision), engineering, and other fields. Geometric algebra and its extension to geometric calculus simplify, unify, and generalize vast areas of mathematics that involve geometric ideas. Geometric algebra is an extension of linear algebra. The treatment of many linear algebra topics is enhanced by geometric algebra, for example, determinants and orthogonal transformations. And geometric algebra does much more, as it incorporates the complex, quaternion, and exterior algebras, among others. Geometric algebra and calculus provide a unified mathematical language for many areas of physics (classical and quantum mechanics, electrodynamics, relativity), computer science (graphics, robotics, computer vision), engineering, and other fields.
  
-  * [[https://sites.google.com/site/winitzki/linalg|Linear Algebra via Exterior Products]] - //Sergei Winitzki//.+  * [[https://sites.google.com/site/winitzki/linalg|Linear Algebra via Exterior Products]] - //Sergei Winitzki//
 The official web site for the free textbook "Linear Algebra via Exterior Products" (2010) by Sergei Winitzki. The official web site for the free textbook "Linear Algebra via Exterior Products" (2010) by Sergei Winitzki.
  
-  * [[http://foundationsofgameenginedev.com|Foundations of Game Engine Development]] - //Eric Lengyel//.+  * [[http://foundationsofgameenginedev.com|Foundations of Game Engine Development]] - //Eric Lengyel//
 Foundations of Game Engine Development is a new book series currently being written by Eric Lengyel. Its four volumes cover the essentials of game engine development in the broad areas of mathematics, rendering, animation, and physics. Foundations of Game Engine Development is a new book series currently being written by Eric Lengyel. Its four volumes cover the essentials of game engine development in the broad areas of mathematics, rendering, animation, and physics.
 +
 +  * [[https://www.mathphysicsbook.com|Mathematics for Physics]] - //Adam Marsh//
 +This book is meant to complement traditional textbooks by covering the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics courses. The idea is to provide an intuitive, visual overview of these mathematical tools, with guiding end goals including but not limited to spinors and gauge theories.
 ==== Journals ==== ==== Journals ====
  
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   * [[https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra|Comparison of vector algebra and geometric algebra]]   * [[https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra|Comparison of vector algebra and geometric algebra]]
   * [[https://en.wikipedia.org/wiki/Clifford_algebra|Clifford algebra]]   * [[https://en.wikipedia.org/wiki/Clifford_algebra|Clifford algebra]]
 +  * [[https://en.wikipedia.org/wiki/Clifford_module|Clifford module]]
 +  * [[https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras|Classification of Clifford algebras]] 
   * [[https://en.wikipedia.org/wiki/Exterior_algebra|Exterior algebra]]   * [[https://en.wikipedia.org/wiki/Exterior_algebra|Exterior algebra]]
   * [[https://en.wikipedia.org/wiki/Grassmann%E2%80%93Cayley_algebra|Grassmann–Cayley algebra]]   * [[https://en.wikipedia.org/wiki/Grassmann%E2%80%93Cayley_algebra|Grassmann–Cayley algebra]]
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   * [[https://en.wikipedia.org/wiki/Vector_space|Vector space]]   * [[https://en.wikipedia.org/wiki/Vector_space|Vector space]]
   * [[https://en.wikipedia.org/wiki/Quaternion|Quaternion]]   * [[https://en.wikipedia.org/wiki/Quaternion|Quaternion]]
 +  * [[https://en.wikipedia.org/wiki/Biquaternion|Biquaternion]]
   * [[https://en.wikipedia.org/wiki/Octonion|Octonion]]   * [[https://en.wikipedia.org/wiki/Octonion|Octonion]]
   * [[https://en.wikipedia.org/wiki/Spinor|Spinor]]   * [[https://en.wikipedia.org/wiki/Spinor|Spinor]]
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   * [[https://en.wikipedia.org/wiki/Multivector|Multivector]]   * [[https://en.wikipedia.org/wiki/Multivector|Multivector]]
   * [[https://en.wikipedia.org/wiki/Paravector|Paravector]]   * [[https://en.wikipedia.org/wiki/Paravector|Paravector]]
 +  * [[https://en.wikipedia.org/wiki/Outermorphism|Outermorphism]]
 +  * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]]
 +  * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]]
  
 ==== Personalities ==== ==== Personalities ====
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   * [[http://www.gdl.cinvestav.mx/~edb|Eduardo Jose Bayro-Corrochano]] - CINVESTAV.   * [[http://www.gdl.cinvestav.mx/~edb|Eduardo Jose Bayro-Corrochano]] - CINVESTAV.
   * [[http://erkenntnis.icu.ac.jp/gcj/software/GAcindy-1.4/GAcindy.htm|Interactive and animated Geometric Algebra with Cinderella]] - //Eckhard Hitzer//, International Christian University.   * [[http://erkenntnis.icu.ac.jp/gcj/software/GAcindy-1.4/GAcindy.htm|Interactive and animated Geometric Algebra with Cinderella]] - //Eckhard Hitzer//, International Christian University.
 +  * [[https://people.well.com/user/billium|William M. Pezzaglia Jr.]] - Department of Physics, Santa Clara University.
   * [[http://cns-alumni.bu.edu/~slehar/Lehar.html|Steven Lehar]] - Boston University.   * [[http://cns-alumni.bu.edu/~slehar/Lehar.html|Steven Lehar]] - Boston University.
   * [[https://www.av8n.com/physics|Physics Documents]] - //John Denker//.   * [[https://www.av8n.com/physics|Physics Documents]] - //John Denker//.
   * [[http://www.garretstar.com|Garret Sobczyk]] - Universidad de las Americas-Puebla.   * [[http://www.garretstar.com|Garret Sobczyk]] - Universidad de las Americas-Puebla.
   * [[http://web.mit.edu/redingtn/www/netadv/biblio3.html|Algebras of Electromagnetics]] - //Perttu P. Puska//, Helsinki University of Technology.   * [[http://web.mit.edu/redingtn/www/netadv/biblio3.html|Algebras of Electromagnetics]] - //Perttu P. Puska//, Helsinki University of Technology.
-  * [[http://www.lomont.org/Math/GeometricAlgebra/Papers.php|Geometric Algebra Papers]] - //Chris Lomont//.+  * [[http://www.lomont.org/math/geometric-algebra/|Geometric Algebra Papers]] - //Chris Lomont//.
   * [[https://people.kth.se/~dogge|Douglas Lundholm]] - Royal Institute of Technology (KTH).   * [[https://people.kth.se/~dogge|Douglas Lundholm]] - Royal Institute of Technology (KTH).
   * [[http://www.iancgbell.clara.net/maths|Maths for (Games) Programmers]] - //Ian Bell//.   * [[http://www.iancgbell.clara.net/maths|Maths for (Games) Programmers]] - //Ian Bell//.
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   * [[http://www.kurtnalty.com|Kurt Nalty]] - Austin Community College.   * [[http://www.kurtnalty.com|Kurt Nalty]] - Austin Community College.
   * [[http://www.terathon.com/lengyel|Eric Lengyel]] - Terathon Software LLC.   * [[http://www.terathon.com/lengyel|Eric Lengyel]] - Terathon Software LLC.
-  * [[http://mdzaharia.eu|Marius Dorian Zaharia]] - University Politehnica of Bucharest. +  * [[https://cs.pub.ro/index.php/people/userprofile/marius_zaharia|Marius Dorian Zaharia]] - University Politehnica of Bucharest. 
-  * [[http://page.math.tu-berlin.de/~gunn|Charles Gunn]] - Technische Universität Berlin.+  * [[http://page.math.tu-berlin.de/~gunn|Charles G. Gunn]] - Institut für Mathematik, Technische Universität Berlin.
   * [[http://vitorpamplona.com|Vitor Pamplona]] - EyeNetra.   * [[http://vitorpamplona.com|Vitor Pamplona]] - EyeNetra.
   * [[https://ga-explorer.netlify.com|Geometric Algebra Explorer]] - //Ahmad Hosny Eid//.   * [[https://ga-explorer.netlify.com|Geometric Algebra Explorer]] - //Ahmad Hosny Eid//.
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   * [[http://www.eleceng.adelaide.edu.au/personal/dabbott|Derek Abbott]] - School of EEE, University of Adelaide.   * [[http://www.eleceng.adelaide.edu.au/personal/dabbott|Derek Abbott]] - School of EEE, University of Adelaide.
   * [[http://www.wolftype.com|Wolftype]] - //Pablo Colapinto//.   * [[http://www.wolftype.com|Wolftype]] - //Pablo Colapinto//.
 +  * [[http://phymatheco.pagesperso-orange.fr|PhyMathEco]] - //Georges Ringeisen//.
   * [[http://810lab.com/md|810 Labs]] - //Alex Arsenovic//.   * [[http://810lab.com/md|810 Labs]] - //Alex Arsenovic//.
   * [[https://enkimute.github.io|Enki's blog]] - //Steven De Keninck//.   * [[https://enkimute.github.io|Enki's blog]] - //Steven De Keninck//.
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   * [[https://www.furey.space|Cohl Furey]] - Trinity Hall, University of Cambridge.   * [[https://www.furey.space|Cohl Furey]] - Trinity Hall, University of Cambridge.
   * [[http://soiguine.com|SOiGUINE Quantum Computing]] - //Alexander Soiguine//.   * [[http://soiguine.com|SOiGUINE Quantum Computing]] - //Alexander Soiguine//.
 +  * [[http://marctenbosch.com|marctenbosch.com]] - //Marc ten Bosch//.
 +  * [[http://www.gary-harper.com|Ripples in Space-Fabric]] - //Gary Harper//.
 +  * [[http://www.martinerikhorn.de|Martin Erik Horn]] - University of Applied Sciences Berlin-Brandenburg.
 +  * [[https://sites.google.com/site/samsilvaunesp|Samuel da Silva]] - Universidade Estadual Paulista.
 +  * [[http://www.siue.edu/~sstaple|George Stacey Staples]] - Department of Mathematics & Statistics, Southern Illinois University Edwardsville.
 +  * [[https://www.richwareham.com|Rich Wareham]] - Department of Engineering, University of Cambridge.
 +  * [[http://ghourabi.net/fadoua|Fadoua Ghourabi]] - Department of Computer Science, Ochanomizu University, Tokyo.
 +  * [[http://www.math.chalmers.se/~rosenan|Andreas Rosén]] - Göteborgs Universitet, Chalmers Tekniska Högskola.
 +  * [[https://www.zatlovac.eu|Václav Zatloukal]] - Department of Physics, Czech Technical University in Prague.
 +  * [[http://www-f1.ijs.si/~pavsic|Matej Pavsic]] - Department of Theoretical Physics, Jožef Stefan Institute Slovenia.
 +  * [[https://rastergraphics.wordpress.com|Rumbo Loxodromico]] - //Mauricio López//.
 ==== Other ==== ==== Other ====
   * [[http://www.williamandlucyclifford.com|William and Lucy Clifford]], A Story of Two Lives   * [[http://www.williamandlucyclifford.com|William and Lucy Clifford]], A Story of Two Lives
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   * [[http://www.nature.com/nature/journal/v48/n1248/abs/048517a0.html|Grassmann's "Ausdehnungslehre"]] - Nature Letters to Editor (1893), //R. W. Genese//   * [[http://www.nature.com/nature/journal/v48/n1248/abs/048517a0.html|Grassmann's "Ausdehnungslehre"]] - Nature Letters to Editor (1893), //R. W. Genese//
-Sir Robert Ball asks why no one has translated the "Ausdehnungslehre" into English. The answer is as regretable as simple—it would not pay. The number of mathematicians who, after the severe courses of the universities, desire to extend their reading is very small. It is something that a respectable few seek to apply what they have already learnt. The first duty of those who direct the studies of the universities is to provide that students may leave in possession of all the best means of future investigation. That fifty years after publication the principles of the "Ausdehnungslehre" should find no place in English mathematical education is indeed astonishing. Half the time given to such a wearisome subject as Lunar Theory would place a student in possession of many of the delightful surprises of Grassmann’s work, and set him thinking for himself. The "Ausdehnungslehre" has won the admiration of too many distinguished mathematicians to remain longer ignored. Clifford said of it: "I may, perhaps, be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science." Useful or not, the work is "a thing of beauty," and no mathematician of taste should pass it by. It is possible, nay, even likely, that its principles may be taught more simply; but the work should be preserved as a classic.+ 
 +Sir Robert Ball asks why no one has translated the "Ausdehnungslehre" into English. The answer is as regretable as simple—it would not pay. The number of mathematicians who, after the severe courses of the universities, desire to extend their reading is very small. It is something that a respectable few seek to apply what they have already learnt. The first duty of those who direct the studies of the universities is to provide that students may leave in possession of all the best means of future investigation. That fifty years after publication the principles of the "Ausdehnungslehre" should find no place in English mathematical education is indeed astonishing. Half the time given to such a wearisome subject as Lunar Theory would place a student in possession of many of the delightful surprises of Grassmann’s work, and set him thinking for himself. The "Ausdehnungslehre" has won the admiration of too many distinguished mathematicians to remain longer ignored. Clifford said of it: "I may, perhaps, be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science." Useful or not, the work is "a thing of beauty," and no mathematician of taste should pass it by. It is possible, nay, even likely, that its principles may be taught more simply; but the work should be preserved as a classic.
  
   * [[https://archive.org/download/jstor-25138478/25138478.pdf|Note on Grassmann's Calculus of Extension]] - Proceedings of the American Academy of Arts and Sciences (1877), //C. S. Peirce//   * [[https://archive.org/download/jstor-25138478/25138478.pdf|Note on Grassmann's Calculus of Extension]] - Proceedings of the American Academy of Arts and Sciences (1877), //C. S. Peirce//
-The last "Mathematische Annalen" contains a paper by H. Grassmann, on the application of his calculus of extension to Mechanics. He adopts the quaternion addition of vectors. But he has two multiplications, internal and external, just as the principles of logic require. (...) The relation of the two multiplications is exceedingly interesting. The system seems to me more suitable to three dimensional space, and also more natural than that of quaternions. The simplification of mechanical formulas is striking, but not more than quaternions would effect, that I see.+ 
 +The last "Mathematische Annalen" contains a paper by H. Grassmann, on the application of his calculus of extension to Mechanics. He adopts the quaternion addition of vectors. But he has two multiplications, internal and external, just as the principles of logic require. (...) The relation of the two multiplications is exceedingly interesting. The system seems to me more suitable to three dimensional space, and also more natural than that of quaternions. The simplification of mechanical formulas is striking, but not more than quaternions would effect, that I see.
 ===== Lecture notes and tutorials ===== ===== Lecture notes and tutorials =====
  
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   * [[http://www.gaalop.de/ga-computing-lecture|Geometric Algebra Computing lecture]] - //Dietmar Hildenbrand//, Technische Universität Darmstadt.   * [[http://www.gaalop.de/ga-computing-lecture|Geometric Algebra Computing lecture]] - //Dietmar Hildenbrand//, Technische Universität Darmstadt.
   * [[https://people.kth.se/~dogge/clifford|Clifford algebra, geometric algebra, and applications]] - //Douglas Lundholm//, Royal Institute of Technology (KTH).   * [[https://people.kth.se/~dogge/clifford|Clifford algebra, geometric algebra, and applications]] - //Douglas Lundholm//, Royal Institute of Technology (KTH).
-  * [[http://www.science.uva.nl/research/ias/ga/publications/CGnA.html|Geometric Algebra: A Computational Framework for Geometrical Applications]] (2002) - //Leo Dorst, Stephen Mann,  Daniel Fontijne//. IEEE Computer Graphics and Applications.+  * Geometric Algebra: A Computational Framework for Geometrical Applications [[https://staff.fnwi.uva.nl/l.dorst/clifford/dorst-mann-I.pdf|Part I]], [[https://staff.fnwi.uva.nl/l.dorst/clifford/dorst-mann-II.pdf|Part II]] (2002) - //Leo Dorst, Stephen Mann,  Daniel Fontijne//. IEEE Computer Graphics and Applications. 
 +  * [[https://staff.fnwi.uva.nl/l.dorst/clifford/CGA3.pdf|Modeling 3D Euclidean Geometry]] (2003) - //Daniel Fontijne, Leo Dorst//. IEEE Computer Graphics and Applications.
   * [[http://www.visgraf.impa.br/courses/ga|Introduction to Geometric Algebra]], Visgraf IMPA - //Leandro A. F. Fernandes, Manuel M. Oliveira//, Instituto de Informática UFRGS.   * [[http://www.visgraf.impa.br/courses/ga|Introduction to Geometric Algebra]], Visgraf IMPA - //Leandro A. F. Fernandes, Manuel M. Oliveira//, Instituto de Informática UFRGS.
   * [[http://isites.harvard.edu/fs/docs/icb.topic1048774.files/clif_mein.pdf|Clifford algebras and Lie groups]] {{ga:clif_mein.pdf|(local)}}- //Eckhard Meinrenken//, University of Toronto.   * [[http://isites.harvard.edu/fs/docs/icb.topic1048774.files/clif_mein.pdf|Clifford algebras and Lie groups]] {{ga:clif_mein.pdf|(local)}}- //Eckhard Meinrenken//, University of Toronto.
   * [[https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009|Applied Geometric Algebra]] (2009) - //László Tisza//, MIT OpenCourseWare.   * [[https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009|Applied Geometric Algebra]] (2009) - //László Tisza//, MIT OpenCourseWare.
 +  * [[https://www.cefns.nau.edu/~schulz/grassmann.pdf|Theory and application of Grassmann Algebra]] (2011) - //William C. Schulz//, Northern Arizona University.
   * [[https://lyseo.edu.ouka.fi/cms7/sites/default/files/geometric_algebra-tl.pdf|What is geometric algebra?]] (2016) - //Teuvo Laurinolli//, Oulun Lyseon.   * [[https://lyseo.edu.ouka.fi/cms7/sites/default/files/geometric_algebra-tl.pdf|What is geometric algebra?]] (2016) - //Teuvo Laurinolli//, Oulun Lyseon.
 +  * [[https://www.zatlovac.eu/lecturenotes/GAIntroLagape.pdf|Geometric Algebra and Calculus: Unified Language for Mathematics and Physics]] (2018) - //Vaclav Zatloukal//, Czech Technical University in Prague.
 +  * [[https://dspace.library.uu.nl/bitstream/handle/1874/383367/IntroductionToGeometricAlgebraV2.pdf|Introduction to Geometric Algebra, a powerful tool for mathematics and physics]] (2019) - //Denis Lamaker//, Universiteit Utrecht.
 +  * [[https://mattferraro.dev/posts/geometric-algebra|What is the Inverse of a Vector?]] (2021) - //Matt Ferraro//.
  
 ===== Videos ===== ===== Videos =====
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   * [[https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K|Geometric Algebra]] playlist - //Mathoma//.   * [[https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K|Geometric Algebra]] playlist - //Mathoma//.
   * [[https://www.youtube.com/playlist?list=PLQ6JJNfj9jD_H3kUopCXkvvGoZqzYOzsV|Geometric Algebra tutorial]] playlist - //Nick Okamoto//.   * [[https://www.youtube.com/playlist?list=PLQ6JJNfj9jD_H3kUopCXkvvGoZqzYOzsV|Geometric Algebra tutorial]] playlist - //Nick Okamoto//.
 +  * [[https://www.youtube.com/watch?v=-6F74TH1i_g&list=PL6oNjS6Kc-nQmqvWjRzLYLk1WlMdFudJa|Exterior Algebra aka Grassmann Algebra]] playlist - //Mathview//.
   * [[https://www.youtube.com/playlist?list=PLB8F2D70E034E9C29|Intro to differential forms]] playlist - //David Metzler//.   * [[https://www.youtube.com/playlist?list=PLB8F2D70E034E9C29|Intro to differential forms]] playlist - //David Metzler//.
   * [[https://www.youtube.com/watch?v=_AaOFCl2ihc|The Vector Algebra War]] - //UniAdel//.   * [[https://www.youtube.com/watch?v=_AaOFCl2ihc|The Vector Algebra War]] - //UniAdel//.
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   * [[https://www.youtube.com/watch?v=WZApQkDBr5o|A Bigger Mathematical Picture for Computer Graphics]] - //Eric Lengyel//.   * [[https://www.youtube.com/watch?v=WZApQkDBr5o|A Bigger Mathematical Picture for Computer Graphics]] - //Eric Lengyel//.
   * [[https://www.youtube.com/watch?v=mz3tk4LRJjc|Introduction to Geometric (Clifford) Algebra]] - //Peter Joot//.   * [[https://www.youtube.com/watch?v=mz3tk4LRJjc|Introduction to Geometric (Clifford) Algebra]] - //Peter Joot//.
-  * [[https://www.youtube.com/watch?v=vOxV9hmXUZU|Applications of Conformal Geometric Algebra to Transmission Line Theory]] - //Alex Arsenovic//.+  * [[https://www.youtube.com/watch?v=vOxV9hmXUZU|Applications of Conformal Geometric Algebra to Transmission Line Theory]] - //Alex Arsenovic (810 Labs)//.
   * [[https://www.youtube.com/watch?v=gLIVCr3duFw|Clifford Algebra, Majorana Particles and the Dirac Equation (by Louis Kauffman)]] - //Institute of Advanced Studies (IAS)//.   * [[https://www.youtube.com/watch?v=gLIVCr3duFw|Clifford Algebra, Majorana Particles and the Dirac Equation (by Louis Kauffman)]] - //Institute of Advanced Studies (IAS)//.
   * [[https://www.youtube.com/watch?v=yLdOvqSIL0I|Introduction to Clifford algebra (by Professor Jose Vargas)]] - //Roger Anderton//.   * [[https://www.youtube.com/watch?v=yLdOvqSIL0I|Introduction to Clifford algebra (by Professor Jose Vargas)]] - //Roger Anderton//.
Line 252: Line 311:
   * [[https://www.youtube.com/watch?v=izV3D3w-CPA|Clifford Algebra (by Y-T Huang)]] - //Yao-Chieh Hu//.   * [[https://www.youtube.com/watch?v=izV3D3w-CPA|Clifford Algebra (by Y-T Huang)]] - //Yao-Chieh Hu//.
   * [[https://www.youtube.com/watch?v=UzEytSmwEpk|Wedge - The Geometric Algebra game]] - //Pedro Alpiarça dos Santos//.   * [[https://www.youtube.com/watch?v=UzEytSmwEpk|Wedge - The Geometric Algebra game]] - //Pedro Alpiarça dos Santos//.
 +  * [[https://www.youtube.com/watch?v=d-4vYtFfet8|Tutorial: Geometric Computing in Computer Graphics using Conformal Geometric Algebra (Japanese)]] - //Kuma Dasu//.
 +  * [[https://www.youtube.com/watch?v=ikCIUzX9myY|Joan Lasenby on Applications of Geometric Algebra in Engineering]] - //Y Combinator//.
 +  * [[https://www.youtube.com/watch?v=Idlv83CxP-8|Let's remove Quaternions from every 3D Engine]] - //Marc ten Bosch//.
 +  * [[https://www.youtube.com/watch?v=hbhxRM_YMv0|Overview of Geometric Algebra]] - //Jack Hanlon//, via //Aaron Murakami//.
 +  * [[https://www.youtube.com/watch?v=P2ZxxoS5YD0|Intro to clifford, a python package for geometric algebra]] - //Alex Arsenovic (810 Labs)//.
 +  * [[https://www.youtube.com/watch?v=QbYao72-V6U|Gamma Matrices and the Clifford Algebra]] - //Pretty Much Physics//.
 +  * [[https://www.youtube.com/watch?v=yG8YKw25f6Y|JMM2018: A Brief introduction to Clifford Algebras]] - //Johannes Familton//.
 +  * [[https://www.youtube.com/watch?v=eQjDN0JQ6-s|JuliaCon 2019: Geometric algebra in Julia with Grassmann.jl]] - //Michael Reed//.
 +  * [[https://skillsmatter.com/skillscasts/13986-geometric-algebra-in-fsharp|Geometric Algebra in F#]] - //Andrew Willshire//.
 +  * [[https://www.youtube.com/watch?v=tX4H_ctggYo|SIGGRAPH 2019: Geometric Algebra for Computer Graphics]] - //Charles Gunn// and //Steven De Keninck//.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjzcQ4eCVAntETNNVD2d5S79|GAME2020 - Geometric Algebra Mini Event]] - //DAE Kortrijk, Belgium//.
 +  * [[https://www.youtube.com/watch?v=60z_hpEAtD8|A Swift Introduction to Geometric Algebra]] and [[https://www.youtube.com/playlist?list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy|From Zero to Geo]] - //sudgylacmoe//.
 +  * [[https://www.youtube.com/watch?v=cKfC2ZBJulg|Projective Geometric Algebra for Paraxial Geometric Optics]] - // Katelyn Spadavecchia//.
 +  * [[https://www.youtube.com/watch?v=11sH9X0OO9Y&list=PLnpuwbuviU2j7OSnZdstP5_g1ejA32bYA|Geometric Algebra Lectures ]] - //Miroslav Josipović//.
 +  * [[https://www.youtube.com/watch?v=HGcBu4TQgRE|Quaternions and Clifford Algebra]] - //Q. J. Ge and Anurag Purwar//, Stony Brook University.
 +  * [[https://www.youtube.com/watch?v=LestlcDk6Iw|Foundations of Geometric Algebra Computing]] - Lecture at ICU Tokyo, //Dietmar Hildenbrand//.
 +  * [[https://www.youtube.com/watch?v=e5D7Bma9Vhw&list=PLxo3PbygE0PLdFFy_2b02JAaUsleFW8py|Geometric Algebra]] - First Course in STEMCstudio, //David Geo Holmes//.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjyWv5wLVV7QfeS_d8pwCPv_|AGACSE2021]] - Selected talks.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc|Plane-based Geometric Algebra Tutorial]] - Presentation at SIBGRAPI 2021, //Steven De Keninck and Leo Dorst//.
 +  * [[https://www.youtube.com/watch?v=hTa_gErsrtM|Geometric Algebra]] - Talk at SIGGRAPH 2022, //Alyn Rockwood and Dietmar Hildenbrand//.
 +  * [[https://www.youtube.com/watch?v=8n6GsKWznfY|Plane Based Geometric Algebra]] - Advanced Computational Applications of GA, //Leo Dorst and Steven De Keninck//.
 +  * [[https://www.youtube.com/watch?v=PGZNYGwsXTw|Why Geometric Algebra Should be in the Standard Linear Algebra Curriculum]] and [[https://www.youtube.com/watch?v=ISKJPmuZkbY|Fun Applications of Geometric Algebra]] - Presentations at [[https://pgadey.ca/seminar/|Parker Glynn-Adey seminars]], //Logan Lim//.
 +  * [[https://www.youtube.com/watch?v=VXziLgMIWf8|Geometric Clifford Algebra Networks and Clifford Neural Layers for PDE Modeling]] - Valence Labs, //Johannes Brandstetter//.
 +  * [[https://www.youtube.com/watch?v=nktgFWLy32U|Spinors for Beginners 11: What is a Clifford Algebra?]] - //eigenchris//.
 +  * [[https://www.youtube.com/watch?v=htYh-Tq7ZBI|Why can't you multiply vectors?]] - Talk at Dutch Game Day 2023, //Freya Holmér//.
 +  * [[https://www.youtube.com/watch?v=1AmeD0Vc8ow|GAME2023 Geometric Algebra Mini Event]] - Livestream.
 +  * [[https://www.youtube.com/watch?v=zgi-13F2Kec|Geometric Algebra: The Prequel]] - at GAME2023, //Steven De Keninck//.
 +  * [[https://www.youtube.com/watch?v=nPIRL-c88_E|Geometric Algebra Transformers: Revolutionizing Geometric Data]] - at Intelligent Systems Conference 2023, //Taco Cohen//.
  
 ===== Computing frameworks ===== ===== Computing frameworks =====
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   * [[http://www.clucalc.info|CLUCalc]] - //Christian Perwass//, Kiel University.   * [[http://www.clucalc.info|CLUCalc]] - //Christian Perwass//, Kiel University.
   * [[http://www.cinderella.de/tiki-index.php|Cinderella]] - The Interactive Geometry Software.   * [[http://www.cinderella.de/tiki-index.php|Cinderella]] - The Interactive Geometry Software.
-  * [[http://www.gaalop.de|Gaalop]] - //Dietmar Hildenbrand//Geometric Algebra Algorithms Optimizer is a software to optimize geometric algebra files. +  * [[http://www.gaalop.de|Gaalop]] - Geometric Algebra Algorithms Optimizer is a software to optimize geometric algebra files, //Dietmar Hildenbrand//
-  * [[https://sourceforge.net/projects/gaalet|Gaalet]] - //Florian Seybold//. Geometric Algebra Algorithms Expression Templates+  * [[https://sourceforge.net/projects/gaalet|Gaalet]] - Geometric Algebra Algorithms Expression Templates, //Florian Seybold//
-  * [[https://sourceforge.net/projects/gaigen|Gaigen]] - //Daniel Fontijne//. Geometric Algebra Implementation Generator+  * [[https://sourceforge.net/projects/gaigen|Gaigen]] - Geometric Algebra Implementation Generator, //Daniel Fontijne//
-  * [[http://glucat.sourceforge.net|GluCat]] - //Paul Leopardi//Library of template classes which model the universal Clifford algebras over the field of real numbers, with arbitrary dimension and arbitrary signature.+  * [[http://glucat.sourceforge.net|GluCat]] - Library of template classes which model the universal Clifford algebras over the field of real numbers, with arbitrary dimension and arbitrary signature, //Paul Leopardi//.
   * [[https://users.aalto.fi/~ppuska/mirror/Lounesto/CLICAL.htm|CLICAL]] - //Pertti Lounesto//, University of Helsinki.   * [[https://users.aalto.fi/~ppuska/mirror/Lounesto/CLICAL.htm|CLICAL]] - //Pertti Lounesto//, University of Helsinki.
   * [[http://versor.mat.ucsb.edu|Versor (libvsr)]] - //Pablo Colapinto//, UC Santa Barbara.   * [[http://versor.mat.ucsb.edu|Versor (libvsr)]] - //Pablo Colapinto//, UC Santa Barbara.
   * [[https://github.com/weshoke/versor.js|versor.js]] - A Javascript port of the Versor geometric algebra library.   * [[https://github.com/weshoke/versor.js|versor.js]] - A Javascript port of the Versor geometric algebra library.
   * [[http://pymbolic.readthedocs.io/en/latest/geometric-algebra.html|Geometric Algebra in pymbolic]] - //Andreas Klöckner//.   * [[http://pymbolic.readthedocs.io/en/latest/geometric-algebra.html|Geometric Algebra in pymbolic]] - //Andreas Klöckner//.
-  * [[http://library.wolfram.com/infocenter/Conferences/6951|Symbolic and Numeric Geometric Algebra]] Mathematica notebook //Terje Vold//.+  * [[https://github.com/grondilu/clifford|clifford: Geometric Algebra in Perl 6]] - //grondilu//
 +  * [[https://github.com/martinbaker/multivector|multivector]] - Code to use with FriCAS, //Martin Baker//. 
 +  * [[http://library.wolfram.com/infocenter/Conferences/6951|Symbolic and Numeric Geometric Algebra]] Mathematica notebook//Terje Vold//. 
 +  * [[http://nklein.com/software/geoma|Geoma]] - //Patrick Stein//.
   * [[https://github.com/andrioni/GeoAlg.jl|Geometric Algebra for Julia]] - //Alessandro Andrioni//.   * [[https://github.com/andrioni/GeoAlg.jl|Geometric Algebra for Julia]] - //Alessandro Andrioni//.
   * [[https://www.geogebra.org/m/qzDtMW2q|Geometric Algebra (Clifford Algebra)]] in GeoGebra - //Jim Smith//.   * [[https://www.geogebra.org/m/qzDtMW2q|Geometric Algebra (Clifford Algebra)]] in GeoGebra - //Jim Smith//.
-  * [[http://www.cs.uu.nl/groups/MG/gallery/CGAP|Conformal Geometric Algebra Package]] for the [[http://www.cgal.org|Computational Geometry Algorithms Library (CGAL)]] //Chaïm Zonnenberg//.+  * [[http://www.cs.uu.nl/groups/MG/gallery/CGAP|Conformal Geometric Algebra Package]] for the [[http://www.cgal.org|Computational Geometry Algorithms Library (CGAL)]]//Chaïm Zonnenberg//.
   * [[https://crypto.stanford.edu/~blynn/haskell/ga.html|All Hail Geometric Algebra!]] - GA explorations in Haskell, //Ben Lynn//.   * [[https://crypto.stanford.edu/~blynn/haskell/ga.html|All Hail Geometric Algebra!]] - GA explorations in Haskell, //Ben Lynn//.
   * [[http://blog.sigfpe.com/2006/08/geometric-algebra-for-free_30.html|Geometric Algebra for Free!]] in Haskell - //Dan Piponi//.   * [[http://blog.sigfpe.com/2006/08/geometric-algebra-for-free_30.html|Geometric Algebra for Free!]] in Haskell - //Dan Piponi//.
   * [[http://clifford-multivector-toolbox.sourceforge.net|Clifford Multivector Toolbox]] for MATLAB - //Steve Sangwine, Eckhard Hitzer//.   * [[http://clifford-multivector-toolbox.sourceforge.net|Clifford Multivector Toolbox]] for MATLAB - //Steve Sangwine, Eckhard Hitzer//.
-  * [[https://github.com/ga-explorer/GMac|GMac]] - Geometric Macro (.NET), //Ahmad Hosny Eid//. +  * [[https://github.com/ga-explorer/GMac|GMac]] and [[https://gmac-guides.netlify.com|GMac Guides]] - Geometric Macro (.NET), //Ahmad Hosny Eid//. 
-  * [[https://github.com/tingelst/game|game]] - Geometric Algebra Multivector Estimation, //Lars Tingelstad//. Framework for estimation of multivectors in geometric algebra with focus on the Euclidean and conformal model.+  * [[https://github.com/tingelst/game|game]] - Geometric Algebra Multivector Estimation framework with focus on the Euclidean and conformal model, //Lars Tingelstad//.
   * [[https://github.com/enkimute/ganja.js|ganja]] - Geometric Algebra for javascript, //Steven De Keninck//.   * [[https://github.com/enkimute/ganja.js|ganja]] - Geometric Algebra for javascript, //Steven De Keninck//.
   * [[https://github.com/jlaragonvera/Geometric-Algebra|CGAlgebra]] - Mathematica package for the 5D Conformal Geometric Algebra, //Jose L. Aragon//.   * [[https://github.com/jlaragonvera/Geometric-Algebra|CGAlgebra]] - Mathematica package for the 5D Conformal Geometric Algebra, //Jose L. Aragon//.
 +  * [[https://github.com/hugohadfield/GAonline|GAonline Tutorial]] and [[https://gaonline.herokuapp.com|App]] - A flask, clifford and threejs/javascript visualiser for (4,1) Conformal Geometric Algebra (CGA), //Hugo Hadfield//.
   * [[https://openga.org|OpenGA]] - Open-source Geometric Algebra, //Wilder Lopes//.   * [[https://openga.org|OpenGA]] - Open-source Geometric Algebra, //Wilder Lopes//.
 +  * [[https://github.com/waivio/cl3|Cl3]] - Cl3 is a Haskell Library implementing standard functions for the Algebra of Physical Space Cl(3,0), //Nathan Waivio//.
 +  * [[https://github.com/reloZid/algeosharp|AlgeoSharp]] - A class library for using conformal geometric algebra in C#, //reloZid//.
 +  * [[https://github.com/stephenathel/gawxm|GAwxM]] - Geometric Algebra using wxMaxima, //Stephen Abbott//.
 +  * [[https://github.com/chakravala/Grassmann.jl|Grassmann.jl]] Julia package - Grassmann-Clifford-Hestenes-Taylor differential geometric algebra of hyper-dual multivector forms, //Dream Scatter//.
 +  * [[https://github.com/pygae|pygae]] - Pythonic Geometric Algebra Enthusiasts at GitHub.
 +  * [[https://www.jeremyong.com/klein|Klein]] - An implementation of 3D Projective Geometric Algebra, //Jeremy Ong//.
 +  * [[http://www.siue.edu/~sstaple/index_files/research.html|CliffMath]] - Clifford algebra computations, including zeon, sym-Clifford, and idem-Clifford subalgebras, //George Stacey Staples//.
 +  * [[https://github.com/Prograf-UFF/TbGAL|TbGAL]] - Tensor-Based Geometric Algebra C++/Python Library, //Eduardo Vera Sousa, Leandro A. F. Fernandes//.
 +  * [[https://github.com/markisus/g3|G3]] - A library for the Geometric Algebra of the Vector Space R^3, //Markisus//.
 +  * [[https://github.com/vincentnozick/garamon|Garamon Generator]] - Geometric Algebra Recursive and Adaptative Monster is a generator of C++ libraries dedicated to Geometric Algebra, //Vincent Nozick, Stephane Breuils//.
 ===== Articles ===== ===== Articles =====
   * [[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf|Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics]] (2002) - //David Hestenes//   * [[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf|Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics]] (2002) - //David Hestenes//
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   * [[https://pdfs.semanticscholar.org/5f79/3ec6aedbd93ee29061407f3fcc3aa2ba5ade.pdf|Clifford Algebra and The Interpretation of Quantum Mechanics]] (1986) - //David Hestenes//   * [[https://pdfs.semanticscholar.org/5f79/3ec6aedbd93ee29061407f3fcc3aa2ba5ade.pdf|Clifford Algebra and The Interpretation of Quantum Mechanics]] (1986) - //David Hestenes//
 The Dirac theory has a hidden geometric structure. This talk traces the conceptual steps taken to uncover that structure and points out significant implications for the interpretation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This implies a geometric interpretation for the generator of electromagnetic gauge transformations as well as for the entire electroweak gauge group of the Weinberg-Salam model. The geometric structure also helps to reveal closer connections to classical theory than hitherto suspected, including exact classical solutions of the Dirac equation. The Dirac theory has a hidden geometric structure. This talk traces the conceptual steps taken to uncover that structure and points out significant implications for the interpretation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This implies a geometric interpretation for the generator of electromagnetic gauge transformations as well as for the entire electroweak gauge group of the Weinberg-Salam model. The geometric structure also helps to reveal closer connections to classical theory than hitherto suspected, including exact classical solutions of the Dirac equation.
 +
 +  * [[https://www.researchgate.net/publication/258944244_Clifford_Algebra_to_Geometric_Calculus_A_Unified_Language_for_Mathematics_and_Physics|Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics]] (1985) - //David Hestenes, Garret Sobczyk, James Marsh//
 +Physics and other applications of mathematics employ a miscellaneous assortment of mathematical tools in ways that contribute to a fragmentation of knowledge. We can do better! Research on the design and use of mathematical systems provides a guide for designing a unified mathematical language for the whole of physics that facilitates learning and enhances insight.The result of developments over several decades is acomprehensive language called Geometric Algebra with wide applications to physics and engineering. This lecture is an introduction to Geometric Algebra with the goal of incorporating it into the math/physics curriculum.
  
   * [[http://www.ejmste.com/pdf-74735-11277?filename=An%20Interview%20with%20David.pdf|An Interview with David Hestenes: His life and achievements]] (2012) - //Mehmet Fatih Taşar, Sedef Canbazoğlu Bilici, Pınar Fettahlıoğlu//   * [[http://www.ejmste.com/pdf-74735-11277?filename=An%20Interview%20with%20David.pdf|An Interview with David Hestenes: His life and achievements]] (2012) - //Mehmet Fatih Taşar, Sedef Canbazoğlu Bilici, Pınar Fettahlıoğlu//
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   * [[http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/DoranThesis.pdf|Geometric Algebra and its Application to Mathematical Physics]] (1994) - //Chris Doran//   * [[http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/DoranThesis.pdf|Geometric Algebra and its Application to Mathematical Physics]] (1994) - //Chris Doran//
 This thesis is an investigation into the properties and applications of Clifford’s geometric algebra. (...) Clifford algebra provides the grammar from which geometric algebra is constructed, but it is only when this grammar is augmented with a number of secondary definitions and concepts that one arrives at a true geometric algebra. In fact, the algebraic properties of a geometric algebra are very simple to understand, they are those of Euclidean vectors, planes and higher-dimensional (hyper)surfaces. It is the computational power brought to the manipulation of these objects that makes geometric algebra interesting and worthy of study. This thesis is an investigation into the properties and applications of Clifford’s geometric algebra. (...) Clifford algebra provides the grammar from which geometric algebra is constructed, but it is only when this grammar is augmented with a number of secondary definitions and concepts that one arrives at a true geometric algebra. In fact, the algebraic properties of a geometric algebra are very simple to understand, they are those of Euclidean vectors, planes and higher-dimensional (hyper)surfaces. It is the computational power brought to the manipulation of these objects that makes geometric algebra interesting and worthy of study.
 +
 +  * [[https://pdfs.semanticscholar.org/5b88/090fe4918e921935a73ce464f0a08e95391b.pdf|Geometric Algebra – Leibnitz' Dream]] (2003) - //Eckhard Hitzer//
 +About 150 years ago, in 1844, the German high school teacher Hermann Grassmann published an ambitious work entitled The Linear Extension Theory, A New Branch of Mathematics. For Grassmann this was indeed The Branch of mathematics, which in his own words "far surpasses" all others. His subsequent work Geometric Algebra won the prize of 45 gold ducats set out by the Princely Jablonowski Society for the recreation and further establishment of the geometric calculus invented by G. W. Leibniz. Grassmann went on to prove the usefulness of his extension theory by applying it to the theory of tides and other phenomena in physics.
  
   * [[http://erkenntnis.icu.ac.jp/gcj/publications/SICEJintro/SICEJintro.pdf|Introduction to William Kingdon Clifford’s geometric algebras paper]] and [[http://erkenntnis.icu.ac.jp/gcj/publications/SICEJintro/SICEJintro_pres.pdf|presentation]] (2011) - //Eckhard Hitzer//   * [[http://erkenntnis.icu.ac.jp/gcj/publications/SICEJintro/SICEJintro.pdf|Introduction to William Kingdon Clifford’s geometric algebras paper]] and [[http://erkenntnis.icu.ac.jp/gcj/publications/SICEJintro/SICEJintro_pres.pdf|presentation]] (2011) - //Eckhard Hitzer//
 Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows e.g. to optimize learning algorithms of Clifford neurons, etc. [[https://arxiv.org/pdf/1306.1660.pdf|arXiv version]] Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows e.g. to optimize learning algorithms of Clifford neurons, etc. [[https://arxiv.org/pdf/1306.1660.pdf|arXiv version]]
 +
 +  * [[https://arxiv.org/pdf/1305.5663v1.pdf|Applications of Clifford's Geometric Algebra]] (2013) - //Eckhard Hitzer, Tohru Nitta, Yasuaki Kuroe//
 +We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly demonstrate the benefit of developing problem solutions in a unified framework for algebra and geometry with the widest possible scope: from quantum computing and electromagnetism to satellite navigation, from neural computing to camera geometry, image processing, robotics and beyond.
 +
 +  * [[https://arxiv.org/pdf/1306.1358.pdf|Geometric operations implemented by conformal geometric algebra neural nodes]] (2013) - //Eckhard Hitzer//
 +(...) Apart from including curved objects, conformal geometric algebra has an elegant unified quaternion like representation for all proper and improper Euclidean transformations, including reflections at spheres, general screw transformations and scaling. Expanding the concepts of real and complex neurons we arrive at the new powerful concept of conformal geometric algebra neurons. These neurons can easily take the above mentioned geometric objects or sets of these objects as inputs and apply a wide range of geometric transformations via the geometric algebra valued weights.
  
   * [[https://www.siam.org/meetings/la09/talks/liesen.pdf|Hermann Graßmann and the Foundations of Linear Algebra]] (2009) - //Jörg Liesen//   * [[https://www.siam.org/meetings/la09/talks/liesen.pdf|Hermann Graßmann and the Foundations of Linear Algebra]] (2009) - //Jörg Liesen//
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   * [[https://arxiv.org/pdf/1306.1007v1.pdf|Euclidean Geometric Objects in the Clifford Geometric Algebra of {Origin, 3-Space, Infinity}]] (2004) - //Eckhard Hitzer//   * [[https://arxiv.org/pdf/1306.1007v1.pdf|Euclidean Geometric Objects in the Clifford Geometric Algebra of {Origin, 3-Space, Infinity}]] (2004) - //Eckhard Hitzer//
 This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and (useful) parametrizations of the 3D Euclidean object models are explicitly demonstrated in order to show how 3D Euclidean information on positions, orientations and radii can be extracted. This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and (useful) parametrizations of the 3D Euclidean object models are explicitly demonstrated in order to show how 3D Euclidean information on positions, orientations and radii can be extracted.
 +
 +  * [[https://www.researchgate.net/publication/228955605_A_brief_introduction_to_Clifford_algebra|A brief introduction to Clifford Algebra]] (2010) - //Silvia Franchini, Giorgio Vassallo, Filippo Sorbello//
 +Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators.
 +
 +  * [[https://vixra.org/pdf/1203.0011v1.pdf|A Very Brief Introduction to Clifford Algebra]] (2012) - //Stephen Crowley//
 +This article distills many of the essential definitions from the very thorough book, Clifford Algebras: An Introduction, by Dr D.J.H. Garling, with some minor additions.
  
   * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf|An Introduction to Geometric Algebra and Calculus]] (2014) - //Alan Bromborsky//   * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf|An Introduction to Geometric Algebra and Calculus]] (2014) - //Alan Bromborsky//
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 This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper. This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper.
  
-  * [[https://sites.math.washington.edu/~morrow/336_17/papers17/josh.pdf|Article Review:  A Survey of Geometric Calculus and Geometric Algebra]] (2017) - //Josh Pollock//+  * [[https://sites.math.washington.edu/~morrow/336_17/papers17/josh.pdf|Article Review: A Survey of Geometric Calculus and Geometric Algebra]] (2017) - //Josh Pollock//
 In his article //A Survey of Geometric Calculus and Geometric Algebra//, Professor Alan Macdonald provides a brief introduction to geometric algebra (GA) and geometric calculus (GC) along with some applications to physics and a brief mention of the related projective and conformal geometric algebras. He only expects the reader to have knowledge of linear algebra and vector calculus. In this review, I hope to whet your appetite for GA and GC by showing some of its important results. In his article //A Survey of Geometric Calculus and Geometric Algebra//, Professor Alan Macdonald provides a brief introduction to geometric algebra (GA) and geometric calculus (GC) along with some applications to physics and a brief mention of the related projective and conformal geometric algebras. He only expects the reader to have knowledge of linear algebra and vector calculus. In this review, I hope to whet your appetite for GA and GC by showing some of its important results.
  
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   * [[https://www.informatik.uni-kiel.de/inf/Sommer/doc/Downloads/Publikationen/GeomComp.pdf|Geometric Computing with Clifford Algebras]] Theoretical Foundations and Applications in Computer Vision and Robotics (2001) - //Gerald Sommer//   * [[https://www.informatik.uni-kiel.de/inf/Sommer/doc/Downloads/Publikationen/GeomComp.pdf|Geometric Computing with Clifford Algebras]] Theoretical Foundations and Applications in Computer Vision and Robotics (2001) - //Gerald Sommer//
-This book presents a collection of contributions concerning the task of solving geometry related problems with suitable algebraic embeddings. It is not only directed at scientists who already discovered the power of Clifford algebras for their field, but also at those scientists who are interested in Clifford algebras and want to see how these can be applied to problems in computer science, signal theory, neural computation, computer vision and robotics. It was therefore tried to keep this book accessible to newcomers to applications of Clifford  algebra while still presenting up to date research and new developments.+This book presents a collection of contributions concerning the task of solving geometry related problems with suitable algebraic embeddings. It is not only directed at scientists who already discovered the power of Clifford algebras for their field, but also at those scientists who are interested in Clifford algebras and want to see how these can be applied to problems in computer science, signal theory, neural computation, computer vision and robotics. It was therefore tried to keep this book accessible to newcomers to applications of Clifford algebra while still presenting up to date research and new developments.
  
   * [[http://www.gaalop.de/dhilden_data/CLUScripts/eg04_tut03.pdf|Geometric Algebra and its Application to Computer Graphics]] (2004) - //D. Hildenbrand, D. Fontijne, C. Perwass, L. Dorst//   * [[http://www.gaalop.de/dhilden_data/CLUScripts/eg04_tut03.pdf|Geometric Algebra and its Application to Computer Graphics]] (2004) - //D. Hildenbrand, D. Fontijne, C. Perwass, L. Dorst//
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   * [[https://arxiv.org/pdf/cs/0203026.pdf|Conformal Geometry, Euclidean Space and Geometric Algebra]] (2002) - //Chris Doran, Anthony Lasenby, Joan Lasenby//   * [[https://arxiv.org/pdf/cs/0203026.pdf|Conformal Geometry, Euclidean Space and Geometric Algebra]] (2002) - //Chris Doran, Anthony Lasenby, Joan Lasenby//
- Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach - the Euclidean distance between points is not handled in a straightforward manner. Here we discuss a solution to this problem, based on conformal geometry. The language of geometric algebra is best suited to exploiting this geometry, as it handles the interior and exterior products in a single, unified framework. A number of applications are discussed, including a compact formula for reflecting a line off a general spherical surface.+Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach - the Euclidean distance between points is not handled in a straightforward manner. Here we discuss a solution to this problem, based on conformal geometry. The language of geometric algebra is best suited to exploiting this geometry, as it handles the interior and exterior products in a single, unified framework. A number of applications are discussed, including a compact formula for reflecting a line off a general spherical surface.
  
   * [[https://arxiv.org/pdf/cs/0310017.pdf|Circle and sphere blending with conformal geometric algebra]] (2003) - //Chris Doran//   * [[https://arxiv.org/pdf/cs/0310017.pdf|Circle and sphere blending with conformal geometric algebra]] (2003) - //Chris Doran//
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   * [[https://arxiv.org/pdf/1205.5935.pdf|Geometric Algebra]] (2012) - //Eric Chisolm//   * [[https://arxiv.org/pdf/1205.5935.pdf|Geometric Algebra]] (2012) - //Eric Chisolm//
-This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands +This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: In addition to scalars and vectors, it defines new objects representing subspaces of any dimension; it defines a product that’s strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It’s actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra".
-on it in two ways: In addition to scalars and vectors, it defines new objects representing subspaces of any dimension; it defines a product that’s strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common +
-plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It’s actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves +
-to be the standard "vector algebra".+
  
   * [[https://arxiv.org/pdf/1011.3698.pdf|Direct Construction of Grassman, Clifford and Geometric Algebras]] (2010) - //A. Cortzen//   * [[https://arxiv.org/pdf/1011.3698.pdf|Direct Construction of Grassman, Clifford and Geometric Algebras]] (2010) - //A. Cortzen//
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   * [[http://link.springer.com/content/pdf/10.1007%2Fs00006-016-0700-z.pdf|Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity]] (2016) - //Anthony Lasenby//   * [[http://link.springer.com/content/pdf/10.1007%2Fs00006-016-0700-z.pdf|Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity]] (2016) - //Anthony Lasenby//
 Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics, physics and engineering. In this contribution, we introduce the Spacetime Algebra (STA), and discuss some of its applications in electromagnetism, quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach. Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics, physics and engineering. In this contribution, we introduce the Spacetime Algebra (STA), and discuss some of its applications in electromagnetism, quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach.
 +
 +  * [[https://arxiv.org/pdf/1504.02906.pdf|Skew ray tracing in a step-index optical fiber using Geometric Algebra]] (2015) - //Angeleene S. Ang, Quirino M. Sugon, Daniel J. McNamara//
 +We used Geometric Algebra to compute the paths of skew rays in a cylindrical, step-index multimode optical fiber. To do this, we used the vector addition form for the law of propagation, the exponential of an imaginary vector form for the law of refraction, and the juxtaposed vector product form for the law of reflection. In particular, the exponential forms of the vector rotations enables us to take advantage of the addition or subtraction of exponential arguments of two rotated vectors in the derivation of the ray tracing invariants in cylindrical and spherical coordinates. We showed that the light rays inside the optical fiber trace a polygonal helical path characterized by three invariants that relate successive reflections inside the fiber: the ray path distance, the difference in axial distances, and the difference in the azimuthal angles. We also rederived the known generalized formula for the numerical aperture for skew rays, which simplifies to the standard form for meridional rays.
  
   * [[https://arxiv.org/ftp/arxiv/papers/1502/1502.02169.pdf|Geometric algebra, qubits, geometric evolution, and all that]] (2015) - //Alexander Soiguine//   * [[https://arxiv.org/ftp/arxiv/papers/1502/1502.02169.pdf|Geometric algebra, qubits, geometric evolution, and all that]] (2015) - //Alexander Soiguine//
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   * [[https://arxiv.org/ftp/arxiv/papers/1805/1805.11953.pdf|The geometric algebra lift of qubits via basic solutions of Maxwell equation]] (2018) - //Alexander Soiguine//   * [[https://arxiv.org/ftp/arxiv/papers/1805/1805.11953.pdf|The geometric algebra lift of qubits via basic solutions of Maxwell equation]] (2018) - //Alexander Soiguine//
- Conventional quantum mechanical qubits can be lifted to states as even three dimensional geometric algebra operators that act on observables. The operators may be implemented via the two types of Maxwell equation solution polarizations. Solution of Maxwell equation in geometric algebra formalism gives g-qubits which are exact lifts of conventional qubits. Therefore, it unambiguously reveals actual meaning of complex parameters of qubits of the commonly accepted Hilbert space quantum mechanics and, particularly, directly demonstrates the option of instant non-locality of states.+Conventional quantum mechanical qubits can be lifted to states as even three dimensional geometric algebra operators that act on observables. The operators may be implemented via the two types of Maxwell equation solution polarizations. Solution of Maxwell equation in geometric algebra formalism gives g-qubits which are exact lifts of conventional qubits. Therefore, it unambiguously reveals actual meaning of complex parameters of qubits of the commonly accepted Hilbert space quantum mechanics and, particularly, directly demonstrates the option of instant non-locality of states
 + 
 +  * [[http://file.scirp.org/pdf/JAMP_2018041914525543.pdf|Polarizations as States and Their Evolution in Geometric Algebra Terms with Variable Complex Plane]] (2018) - //Alexander Soiguine// 
 +Recently suggested scheme of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors. 
 + 
 +  * [[https://arxiv.org/pdf/1807.08603|State/observable interactions using basic geometric algebra solutions of the Maxwell equation]] (2018) - //Alexander Soiguine// 
 +Maxwell equation in geometric algebra formalism with equally weighted basic solutions is subjected to continuously acting Clifford translation. The received states, operators acting on observables, are analyzed with different values of the Clifford translation time factor and through the observable measurement results.
  
   * [[https://arxiv.org/pdf/physics/0410035.pdf|The null subspace of G(4,1) as source of the main physical theories]] (2004) - //Jose B. Almeida//   * [[https://arxiv.org/pdf/physics/0410035.pdf|The null subspace of G(4,1) as source of the main physical theories]] (2004) - //Jose B. Almeida//
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   * [[https://arxiv.org/pdf/math-ph/0403040.pdf|The Construction of Spinors in Geometric Algebra]] (2008) - //Matthew R. Francis, Arthur Kosowsky//   * [[https://arxiv.org/pdf/math-ph/0403040.pdf|The Construction of Spinors in Geometric Algebra]] (2008) - //Matthew R. Francis, Arthur Kosowsky//
 The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U(1), SU(2), and SL(2,C) spinors. The physical observables in Schr¨odinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches. The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U(1), SU(2), and SL(2,C) spinors. The physical observables in Schr¨odinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches.
 +
 +  * [[https://arxiv.org/pdf/1312.3824.pdf|An introduction to spinors]] (2013) - //Andrew M. Steane//
 +We introduce spinors, at a level appropriate for an undergraduate or first year graduate course on relativity, astrophysics or particle physics. The treatment assumes very little mathematical knowledge (mainly just vector analysis and some idea of what a group is). The SU(2)-SO(3) homomorphism is presented in detail. Lorentz transformation, chirality, and the spinor Minkowski metric are introduced. Applications to electromagnetism, parity violation, and to Dirac spinors are presented. A classical form of the Dirac equation is obtained, and the (quantum) prediction that g=2 for Dirac particles is presented.
  
   * [[https://www.researchgate.net/profile/Asao_Arai/publication/226563528_Characterization_of_Anticommutativity_of_Self-Adjoint_Operators_in_Connection_with_Clifford_Algebra_and_Applications/links/02bfe50fde05763ec8000000/Characterization-of-Anticommutativity-of-Self-Adjoint-Operators-in-Connection-with-Clifford-Algebra-and-Applications.pdf|Characterization of Anticommutativity of Self-Adjoint Operators in Connection with Clifford Algebra and Applications]] (1993) - //Asao Arai//   * [[https://www.researchgate.net/profile/Asao_Arai/publication/226563528_Characterization_of_Anticommutativity_of_Self-Adjoint_Operators_in_Connection_with_Clifford_Algebra_and_Applications/links/02bfe50fde05763ec8000000/Characterization-of-Anticommutativity-of-Self-Adjoint-Operators-in-Connection-with-Clifford-Algebra-and-Applications.pdf|Characterization of Anticommutativity of Self-Adjoint Operators in Connection with Clifford Algebra and Applications]] (1993) - //Asao Arai//
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   * [[http://www.reed.edu/physics/faculty/wheeler/documents/Electrodynamics/Miscellaneous%20Essays/Maxwellian%20Theories.pdf|Theories of Maxwellian Design]] (1998) - //Nicholas Wheeler//   * [[http://www.reed.edu/physics/faculty/wheeler/documents/Electrodynamics/Miscellaneous%20Essays/Maxwellian%20Theories.pdf|Theories of Maxwellian Design]] (1998) - //Nicholas Wheeler//
-(...) It is my strong feeling that it is time—long past time—to bring that +(...) It is my strong feeling that it is time—long past time—to bring that unfortunate state of affairs to an end. It is wrong to suppose that one need be a state-of-the-art differential geometer to make effective use of the exterior calculus. The formalism imposes demands certainly not greater than those of tensor analysis (from which it arises as a powerful special case), and in practical applications does its work even more simply and swiftly than vector analysis. And, as it is my intention here to demonstrate, it does pass Kelvin’s test, does put us in position to say new things about some moderately interesting new questions, to say things that are quite beyond the reach of vector analysis, and which if formulated in the language of tensor analysis would entail such a “debouch d'indices” as to remain almost certainly unsaid.
-unfortunate state of affairs to an end. It is wrong to suppose that one need +
-be a state-of-the-art differential geometer to make effective use of the exterior +
-calculus. The formalism imposes demands certainly not greater than those of +
-tensor analysis (from which it arises as a powerful special case), and in practical +
-applications does its work even more simply and swiftly than vector analysis. +
-And, as it is my intention here to demonstrate, it does pass Kelvin’s test, does +
-put us in position to say new things about some moderately interesting new +
-questions, to say things that are quite beyond the reach of vector analysis, +
-and which if formulated in the language of tensor analysis would entail such a +
-“debouch d'indices” as to remain almost certainly unsaid.+
  
   * [[http://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous Math/Clifford Algebra.pdf|Transformational principles latent in the theory of Clifford Algebras]] (2003) - //Nicholas Wheeler//   * [[http://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous Math/Clifford Algebra.pdf|Transformational principles latent in the theory of Clifford Algebras]] (2003) - //Nicholas Wheeler//
-My purpose in this informal material will be to refresh/consolidate +My purpose in this informal material will be to refresh/consolidate my own thought concerning a subject that has engaged my attention from time to time over the years, but not recently. I do so because aspects of the subject have become the focus of a thesis effort by Nakul Shankar, whose awkward situation is that he is going to have to change horses in midstream: I will direct the first phase of his project, but will be obliged to pass the baton to Tom Wieting & Darrell Schroeter at mid-year. A large part of my effort here, therefore, will be directed to the the establishment of some degree of notational and conceptual commonality, and to construction of a clear statement of my own motivating interests in this area.
-my own thought concerning a subject that has engaged my attention from time +
-to time over the years, but not recently. I do so because aspects of the subject +
-have become the focus of a thesis effort by Nakul Shankar, whose awkward +
-situation is that he is going to have to change horses in midstream: I will +
-direct the first phase of his project, but will be obliged to pass the baton to +
-Tom Wieting & Darrell Schroeter at mid-year. A large part of my effort here, +
-therefore, will be directed to the the establishment of some degree of notational +
-and conceptual commonality, and to construction of a clear statement of my +
-own motivating interests in this area.+
  
   * [[https://cs.uwaterloo.ca/research/tr/1999/27/CS-99-27.pdf|The Making of a Geometric Algebra Package in Matlab]] (1999) - //Stephen Mann, Leo Dorst, Tim Bouma//   * [[https://cs.uwaterloo.ca/research/tr/1999/27/CS-99-27.pdf|The Making of a Geometric Algebra Package in Matlab]] (1999) - //Stephen Mann, Leo Dorst, Tim Bouma//
 In this paper, we describe our development of GABLE, a Matlab implementation of the Geometric Algebra based on Cl(p,q) (where p + q = 3) and intended for tutorial purposes. Of particular note are the matrix representation of geometric objects, effective algorithms for this geometry (inversion, meet and join), and issues in efficiency and numerics. In this paper, we describe our development of GABLE, a Matlab implementation of the Geometric Algebra based on Cl(p,q) (where p + q = 3) and intended for tutorial purposes. Of particular note are the matrix representation of geometric objects, effective algorithms for this geometry (inversion, meet and join), and issues in efficiency and numerics.
  
-  * [[https://www.staff.science.uu.nl/~kreve101/asci/GAraytracer.pdf|Performance and elegance of five models of +  * [[https://www.staff.science.uu.nl/~kreve101/asci/GAraytracer.pdf|Performance and elegance of five models of 3D Euclidean geometry in a ray tracing application]] (2003) - //Daniel Fontijne, Leo Dorst//
-3D Euclidean geometry in a ray tracing application]] (2003) - //Daniel Fontijne, Leo Dorst//+
 Computations of 3D Euclidean geometry can be performed using various computational models of different effectiveness. In this paper we compare five alternatives: 3D linear algebra, 3D geometric algebra, a mix of 4D homogeneous coordinates and Plucker coordinates, a 4D homogeneous model using geometric algebra, and the 5D conformal model using geometric algebra. Computations of 3D Euclidean geometry can be performed using various computational models of different effectiveness. In this paper we compare five alternatives: 3D linear algebra, 3D geometric algebra, a mix of 4D homogeneous coordinates and Plucker coordinates, a 4D homogeneous model using geometric algebra, and the 5D conformal model using geometric algebra.
  
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 We express Maxwell's equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the Electromagnetic field. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. The decoupling of the equations into the wave equation is very simple and natural. The divergence of the stress energy tensor gives the Lorentz Law in a very natural way. We compare this approach to the related descriptions of Maxwell’s equations by biquaternions and Clifford algebras. We express Maxwell's equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the Electromagnetic field. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. The decoupling of the equations into the wave equation is very simple and natural. The divergence of the stress energy tensor gives the Lorentz Law in a very natural way. We compare this approach to the related descriptions of Maxwell’s equations by biquaternions and Clifford algebras.
  
-  * [[http://ieeexplore.ieee.org/iel7/5/6879517/06876131.pdf|Geometric Algebra for Electrical +  * [[http://ieeexplore.ieee.org/iel7/5/6879517/06876131.pdf|Geometric Algebra for Electrical and Electronic Engineers]] (2014) - //James M. Chappell et al.//
-and Electronic Engineers]] (2014) - //James M. Chappell et al.//+
 In this paper, we explicate the suggested benefits of Clifford's geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases. In this paper, we explicate the suggested benefits of Clifford's geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.
  
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 Geometric algebra has been proved to be a powerful mathematical language for robot vision. We give an overview of some research results in two different areas of robot vision. These are signal theory in the multidimensional case and knowledge based neural computing. In both areas a considerable extension of the modeling has been achieved by applying geometric algebra. Geometric algebra has been proved to be a powerful mathematical language for robot vision. We give an overview of some research results in two different areas of robot vision. These are signal theory in the multidimensional case and knowledge based neural computing. In both areas a considerable extension of the modeling has been achieved by applying geometric algebra.
  
-  * [[https://www.informatik.uni-kiel.de/inf/Sommer/doc/Publications/chp/perwass_thesis00.pdf|Applications of +  * [[https://www.informatik.uni-kiel.de/inf/Sommer/doc/Publications/chp/perwass_thesis00.pdf|Applications of Geometric Algebra in Computer Vision]] (2000) - //Christian Perwass//
-Geometric Algebra in Computer Vision]] (2000) - //Christian Perwass//+
 (...) We discuss multiple view tensors from a geometric point of view. We show that in this way multiple view tensors can be expressed in a unified way, and that constraints on them can be found from simple geometric considerations. In the last part of this chapter we discuss projective reconstructions from trifocal tensors. We find that the consistency of a trifocal tensor has no particular influence on the quality of reconstruction. (...) We discuss multiple view tensors from a geometric point of view. We show that in this way multiple view tensors can be expressed in a unified way, and that constraints on them can be found from simple geometric considerations. In the last part of this chapter we discuss projective reconstructions from trifocal tensors. We find that the consistency of a trifocal tensor has no particular influence on the quality of reconstruction.
  
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   * [[https://arxiv.org/pdf/1411.6502.pdf|Geometric Algebras for Euclidean Geometry]] (2016) - //Charles G. Gunn//   * [[https://arxiv.org/pdf/1411.6502.pdf|Geometric Algebras for Euclidean Geometry]] (2016) - //Charles G. Gunn//
 The discussion of how to apply geometric algebra to euclidean n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from 19th century mathematics. We then introduce the dual projectivized Clifford algebra P(R∗_n,0,1) (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.  The discussion of how to apply geometric algebra to euclidean n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from 19th century mathematics. We then introduce the dual projectivized Clifford algebra P(R∗_n,0,1) (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA. 
 +
 +  * [[https://arxiv.org/pdf/1501.06511.pdf|Doing euclidean plane geometry using projective geometric algebra]] (2016) - //Charles G. Gunn//
 +The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief review of PGA, the article focuses on P(R∗_2,0,1), the PGA for euclidean plane geometry. It first explores the geometric product involving pairs and triples of basic elements (points and lines), establishing a wealth of fundamental metric and non-metric properties. It then applies the algebra to a variety of familiar topics in plane euclidean geometry and shows that it compares favorably with other approaches in regard to completeness, compactness, practicality, and elegance. The seamless integration of euclidean and ideal (aka infinite) elements forms an essential and novel feature of the treatment.
  
   * [[http://www.gaalop.de/wp-content/uploads/CGI_CGA_Paper.pdf|An inclusive Conformal Geometric Algebra GPU animation interpolation and deformation algorithm]] (2016)   * [[http://www.gaalop.de/wp-content/uploads/CGI_CGA_Paper.pdf|An inclusive Conformal Geometric Algebra GPU animation interpolation and deformation algorithm]] (2016)
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 This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel presentation of several topics of the differential geometry of (smooth) manifolds of arbitrary global topology. A key tool for the development of our program is the mastering of the euclidean geometrical algebra of multivectors that is detailed in the present paper. This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel presentation of several topics of the differential geometry of (smooth) manifolds of arbitrary global topology. A key tool for the development of our program is the mastering of the euclidean geometrical algebra of multivectors that is detailed in the present paper.
  
-  * [[https://arxiv.org/pdf/math-ph/0212043.pdf|Euclidean Clifford Algebra]] (2001) - //V. V. Fernandez, +  * [[https://arxiv.org/pdf/math-ph/0212043.pdf|Euclidean Clifford Algebra]] (2001) - //V. V. Fernandez, A. M. Moya, W. A. Rodrigues Jr.//
-A. M. Moya, W. A. Rodrigues Jr.//+
 In this paper we introduce the concept of euclidean Clifford algebra Cl(V,GE) for a given euclidean structure on V , i.e., a pair (V,GE) where GE is a euclidean metric for V (also called an euclidean scalar product). In this paper we introduce the concept of euclidean Clifford algebra Cl(V,GE) for a given euclidean structure on V , i.e., a pair (V,GE) where GE is a euclidean metric for V (also called an euclidean scalar product).
  
-  * [[https://arxiv.org/pdf/math-ph/0212048.pdf|Metric Tensor Vs. Metric Extensor]] (2001) - //V. V. Fernandez, +  * [[https://arxiv.org/pdf/math-ph/0212048.pdf|Metric Tensor Vs. Metric Extensor]] (2001) - //V. V. Fernandez, A. M. Moya, W. A. Rodrigues Jr.//
-A. M. Moya, W. A. Rodrigues Jr.//+
 In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of tensors and extensors. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors do not have inverses. In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of tensors and extensors. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors do not have inverses.
  
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   * [[https://arxiv.org/pdf/math-ph/0302007.pdf|Lagrangian Formalism for Multiform Fields on Minkowski Spacetime]] (2016) - //V. V. Fernandez, A. M. Moya, W. A. Rodrigues Jr.//   * [[https://arxiv.org/pdf/math-ph/0302007.pdf|Lagrangian Formalism for Multiform Fields on Minkowski Spacetime]] (2016) - //V. V. Fernandez, A. M. Moya, W. A. Rodrigues Jr.//
-We present an introduction to the mathematical theory of the Lagrangian formalism for multiform fields on Minkowski spacetime based on the multiform and extensor calculus. Our formulation gives a unified +We present an introduction to the mathematical theory of the Lagrangian formalism for multiform fields on Minkowski spacetime based on the multiform and extensor calculus. Our formulation gives a unified mathematical description for the main relativistic field theories including the gravitational field.
-mathematical description for the main relativistic field theories including the gravitational field.+
  
   * [[https://arxiv.org/pdf/math/0501558.pdf|Extensors in Geometric Algebra]] (2008) - //V. V. Fernandez, A. M. Moya, W. A. Rodrigues Jr.//   * [[https://arxiv.org/pdf/math/0501558.pdf|Extensors in Geometric Algebra]] (2008) - //V. V. Fernandez, A. M. Moya, W. A. Rodrigues Jr.//
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 A tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics. A tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics.
  
-  * [[https://pure.uva.nl/ws/files/4375498/52687_fontijne.pdf|Efficient Implementation of Geometric Algebra]]  (2007) - //Daniel Fontijne//+  * [[https://pure.uva.nl/ws/files/4375498/52687_fontijne.pdf|Efficient Implementation of Geometric Algebra]] (2007) - //Daniel Fontijne//
 This thesis addresses the computational and implementational aspects of geometric algebra, and shows that its mathematical promise can be made into programming reality: geometric algebra provides a modular, structured specification language for geometry whose implementations can be automatically generated, leading to an efficiency that is competitive with the (hand-) optimized code based on the traditional linear algebra approach. This thesis addresses the computational and implementational aspects of geometric algebra, and shows that its mathematical promise can be made into programming reality: geometric algebra provides a modular, structured specification language for geometry whose implementations can be automatically generated, leading to an efficiency that is competitive with the (hand-) optimized code based on the traditional linear algebra approach.
  
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   * [[http://www.cosmo.fiz.univ.szczecin.pl/pliki/grasscosmofun/talks/A_Lasenby/anl_szczecin_09.pdf|Grassmann, Geometric Algebra and Cosmology]] (2009) - //Anthony Lasenby//   * [[http://www.cosmo.fiz.univ.szczecin.pl/pliki/grasscosmofun/talks/A_Lasenby/anl_szczecin_09.pdf|Grassmann, Geometric Algebra and Cosmology]] (2009) - //Anthony Lasenby//
-Will discuss some aspects along the route from Grassmann to Cosmology via Geometric Algebra: A short introduction to Geometric Algebra and an interesting new version — Conformal Geometric Algebra (CGA). +Will discuss some aspects along the route from Grassmann to Cosmology via Geometric Algebra: A short introduction to Geometric Algebra and an interesting new version — Conformal Geometric Algebra (CGA). Brief advertisement for Gauge Theory Gravity, and a recent extension to scale invariance. Application of CGA to de Sitter space and a novel slightly closed universe model CGA and Bianchi models, including a non-singular Bianchi IX universe. How the Cosmic Microwave Background constrains these, and some other areas in fundamental physics.
-Brief advertisement for Gauge Theory Gravity, and a recent extension to scale invariance. Application of CGA to de Sitter space and a novel slightly closed universe model CGA and Bianchi models, including a non-singular Bianchi IX universe. How the Cosmic Microwave Background constrains these, and some other areas in fundamental physics.+
  
-  * [[http://www.gaalop.de/dhilden_data/ICCA7.pdf|Inverse kinematics computation in computer graphics and +  * [[http://www.gaalop.de/dhilden_data/ICCA7.pdf|Inverse kinematics computation in computer graphics and robotics using conformal geometric algebra]] (2005) - //Dietmar Hildenbrand, Julio Zamora, Eduardo Bayro-Corrochano//
-robotics using conformal geometric algebra]] (2005) - //Dietmar Hildenbrand, Julio Zamora, Eduardo Bayro-Corrochano//+
 We focus on inverse kinematics applications in computer graphics and robotics based on Conformal Geometric Algebra. Here, geometric objects like spheres and circles that are often needed in inverse kinematics algorithms are simply represented by algebraic objects. We present algorithms for the inverse kinematics of a human arm like kinematic chain and for the grasping of robots and virtual humans. The main benefits of using geometric algebra in the virtual reality software Avalon are the easy, compact and geometrically intuitive formulation of the algorithms and the immediate computation of quaternions. We focus on inverse kinematics applications in computer graphics and robotics based on Conformal Geometric Algebra. Here, geometric objects like spheres and circles that are often needed in inverse kinematics algorithms are simply represented by algebraic objects. We present algorithms for the inverse kinematics of a human arm like kinematic chain and for the grasping of robots and virtual humans. The main benefits of using geometric algebra in the virtual reality software Avalon are the easy, compact and geometrically intuitive formulation of the algorithms and the immediate computation of quaternions.
  
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 In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method.
  
-  * [[http://www.mic-journal.no/PDF/2016/MIC-2016-1-6.pdf|Inverse Kinematics for Industrial Robots using +  * [[http://www.mic-journal.no/PDF/2016/MIC-2016-1-6.pdf|Inverse Kinematics for Industrial Robots using Conformal Geometric Algebra]] (2016) - //A. Kleppe, O. Egeland//
-Conformal Geometric Algebra]] (2016) - //A. Kleppe, O. Egeland//+
 This paper shows how the recently developed formulation of conformal geometric algebra can be used for analytic inverse kinematics of two six-link industrial manipulators with revolute joints. The paper demonstrates that the solution of the inverse kinematics in this framework relies on the intersection of geometric objects like lines, circles, planes and spheres, which provides the developer with valuable geometric intuition about the problem. It is believed that this will be very useful for new robot geometries and other mechanisms like cranes and topside drilling equipment. The paper extends previous results on inverse kinematics using conformal geometric algebra by providing consistent solutions for the joint angles for the different configurations depending on shoulder left or right, elbow up or down, and wrist flipped or not. Moreover, it is shown how to relate the solution to the Denavit-Hartenberg parameters of the robot. This paper shows how the recently developed formulation of conformal geometric algebra can be used for analytic inverse kinematics of two six-link industrial manipulators with revolute joints. The paper demonstrates that the solution of the inverse kinematics in this framework relies on the intersection of geometric objects like lines, circles, planes and spheres, which provides the developer with valuable geometric intuition about the problem. It is believed that this will be very useful for new robot geometries and other mechanisms like cranes and topside drilling equipment. The paper extends previous results on inverse kinematics using conformal geometric algebra by providing consistent solutions for the joint angles for the different configurations depending on shoulder left or right, elbow up or down, and wrist flipped or not. Moreover, it is shown how to relate the solution to the Denavit-Hartenberg parameters of the robot.
  
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 (...) In this chapter we are concerned with the extension of the representation capabilities of the dual numbers, particularly using the motor algebra beside the point and line representation we are able to model the motion of planes. This widens up the possibilities for the modelling of the motion of the basic geometric objects referred to frames attached to the robot manipulator which according to the circumstances simplify the complexity of the problem preserving the underlying geometry. After giving the modelling of prismatic and revolute transformations of a robot manipulator using points, lines and planes we solve the direct and inverse kinematics of robot manipulators. Using the motion of points, lines and planes in terms of motors we present constraints for a simple grasping task. The chapter shows clearly the advantages of the use of representations in motor algebra for solving problems related to robot manipulators. (...) In this chapter we are concerned with the extension of the representation capabilities of the dual numbers, particularly using the motor algebra beside the point and line representation we are able to model the motion of planes. This widens up the possibilities for the modelling of the motion of the basic geometric objects referred to frames attached to the robot manipulator which according to the circumstances simplify the complexity of the problem preserving the underlying geometry. After giving the modelling of prismatic and revolute transformations of a robot manipulator using points, lines and planes we solve the direct and inverse kinematics of robot manipulators. Using the motion of points, lines and planes in terms of motors we present constraints for a simple grasping task. The chapter shows clearly the advantages of the use of representations in motor algebra for solving problems related to robot manipulators.
  
-  * [[http://www.scielo.org.mx/pdf/cys/v19n3/v19n3a6.pdf|Modeling and Pose Control of Robotic Manipulators and Legs +  * [[http://www.scielo.org.mx/pdf/cys/v19n3/v19n3a6.pdf|Modeling and Pose Control of Robotic Manipulators and Legs using Conformal Geometric Algebra]] (2015) - //Oscar Carbajal-Espinosa et al// 
-using Conformal Geometric Algebra]] (2015) - //Oscar Carbajal-Espinosa et al// +Controlling the pose of a manipulator involves finding the correct configuration of the robot’s elements to move the end effector to a desired position and orientation. In order to find the geometric relationships between the elements of a robot manipulator, it is necessary to define the kinematics of the robot. We present a synthesis of the kinematical model of the pose for this type of robot using the conformal geometric algebra framework. In addition, two controllers are developed, one for the position tracking problem and another for the orientation tracking problem, both using an error feedback controller. The stability analysis is carried out for both controllers, and their application to a 6-DOF serial manipulator and the legs of a biped robot are presented. By proposing the error feedback and Lyapunov functions in terms of geometric algebra, we are opening a new venue of research in control of manipulators and robot legs that involves the use of geometric primitives, such as lines, circles, planes, spheres.
-Controlling the pose of a manipulator involves finding the correct configuration of the robot’s elements to move the end effector to a desired position and orientation. In  order  to  find  the  geometric  relationships between the elements of a robot manipulator, it is necessary to define the kinematics of the robot. We present a  synthesis  of the kinematical model of the pose for this type of robot using the conformal geometric algebra framework. In addition,  two controllers are developed, one for the position tracking problem and another for the orientation tracking problem, both using an error feedback controller. The stability analysis is carried out for both controllers, and their application to a 6-DOF serial manipulator and the legs of a biped robot are presented. By proposing the error feedback and Lyapunov functions in  terms  of  geometric  algebra,  we  are  opening   new venue of research in control of manipulators and robot legs that involves the use of geometric primitives, such as lines, circles, planes, spheres.+
  
   * [[https://www.researchgate.net/profile/Leo_Dorst/publication/254901215_Competitive_runtime_performance_for_inverse_kinematics_algorithms_using_conformal_geometric_algebra/links/5444bfb20cf2a76a3ccd81cd.pdf|Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra]] (2006) - //Dietmar Hildenbrand, Daniel Fontijne, Yusheng Wang, Marc Alexa, Leo Dorst//   * [[https://www.researchgate.net/profile/Leo_Dorst/publication/254901215_Competitive_runtime_performance_for_inverse_kinematics_algorithms_using_conformal_geometric_algebra/links/5444bfb20cf2a76a3ccd81cd.pdf|Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra]] (2006) - //Dietmar Hildenbrand, Daniel Fontijne, Yusheng Wang, Marc Alexa, Leo Dorst//
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 This paper describes advancement in color edge detection, using a dedicated Geometric Algebra (GA) co-processor implemented on an Application Specific Integrated Circuit (ASIC). GA provides a rich set of geometric operations, giving the advantage that many signal and image processing operations become straightforward and the algorithms intuitive to design. The use of GA allows images to be represented with the three R, G, B color channels defined as a single entity, rather than separate quantities. A novel custom ASIC is proposed and fabricated that directly targets GA operations and results in significant performance improvement for color edge detection. This paper describes advancement in color edge detection, using a dedicated Geometric Algebra (GA) co-processor implemented on an Application Specific Integrated Circuit (ASIC). GA provides a rich set of geometric operations, giving the advantage that many signal and image processing operations become straightforward and the algorithms intuitive to design. The use of GA allows images to be represented with the three R, G, B color channels defined as a single entity, rather than separate quantities. A novel custom ASIC is proposed and fabricated that directly targets GA operations and results in significant performance improvement for color edge detection.
  
-  * [[https://www.esa.informatik.tu-darmstadt.de/twiki/pub/Staff/AndreasKochPublications/090305SpringerChapter.pdf|Gaalop - High Performance Parallel Computing based on Conformal Geometric Algebra]] (2009) - //Dietmar Hildenbrand, Joachim Pitt, Andreas Koch//+  * [[http://www.gaalop.de/dhilden_data/SpringerHildKochPitt.pdf|Gaalop - High Performance Parallel Computing based on Conformal Geometric Algebra]] (2009) - //Dietmar Hildenbrand, Joachim Pitt, Andreas Koch//
 We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computing based on conformal geometric algebra. The main goal of Gaalop is to realize implementations that are most likely faster than conventional solutions. In order to achieve this goal, our focus is on parallel target platforms like FPGA (field-programmable gate arrays) or the CUDA technology from NVIDIA. We describe the concepts, the current status, as well as the future perspectives of Gaalop dealing with optimized software implementations, hardware implementations as well as mixed solutions. An inverse kinematics algorithm of a humanoid robot is described as an example. We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computing based on conformal geometric algebra. The main goal of Gaalop is to realize implementations that are most likely faster than conventional solutions. In order to achieve this goal, our focus is on parallel target platforms like FPGA (field-programmable gate arrays) or the CUDA technology from NVIDIA. We describe the concepts, the current status, as well as the future perspectives of Gaalop dealing with optimized software implementations, hardware implementations as well as mixed solutions. An inverse kinematics algorithm of a humanoid robot is described as an example.
  
   * [[http://www.gaalop.de/wp-content/uploads/Masterthesis.pdf|Geometric Algebra enhanced Precompiler for C++ and OpenCL]] (2012) - //Patrick Charrier//   * [[http://www.gaalop.de/wp-content/uploads/Masterthesis.pdf|Geometric Algebra enhanced Precompiler for C++ and OpenCL]] (2012) - //Patrick Charrier//
-From a programmer's perspective, the integration of GA directly into C++, OpenCL, and CUDA, yields a high +From a programmer's perspective, the integration of GA directly into C++, OpenCL, and CUDA, yields a high level of intuitiveness. Coupled with a highly efficient generative software tool like Gaalop in the background, an integration sets new standards to GA-powered software development. An advanced integration itself including other comforts, and to make GA-usage available to a broad audience, is the purpose of this work.
-level of intuitiveness. Coupled with a highly efficient generative software tool like Gaalop in the background, an integration sets new standards to GA-powered software development. An advanced integration itself including other comforts, and to make GA-usage available to a broad audience, is the purpose of this work.+
  
   * [[https://mat-web.upc.edu/people/sebastia.xambo/agacse2015/3641572286-EP/150729%20AGACSE%202015%20Early%20Proceedings.pdf|Early Proceedings of the AGACSE 2015 Conference]] (2015)   * [[https://mat-web.upc.edu/people/sebastia.xambo/agacse2015/3641572286-EP/150729%20AGACSE%202015%20Early%20Proceedings.pdf|Early Proceedings of the AGACSE 2015 Conference]] (2015)
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   * [[http://www.osti.gov/scitech/servlets/purl/468622|Using Geometric Algebra to Understand Pattern Rotations in Multiple Mirror Optical Systems]] (1997) - //Jack Hanlon, Hans Ziock//   * [[http://www.osti.gov/scitech/servlets/purl/468622|Using Geometric Algebra to Understand Pattern Rotations in Multiple Mirror Optical Systems]] (1997) - //Jack Hanlon, Hans Ziock//
-Geometric Algebra (GA) is a new formulation of Clifford Algebra that includes vector analysis without +Geometric Algebra (GA) is a new formulation of Clifford Algebra that includes vector analysis without notation changes. Most applications of GA have been in theoretical physics, but GA is also a very good analysis tool for engineering. As an example, we use GA to study pattern rotation in optical systems with multiple mirror reflections. The common ways to analyze pattern rotations are to use rotation matrices or optical ray trace codes, but these are often inconvenient. We use GA to develop a simple expression for pattern rotation that is useful for designing or tolerancing pattern rotations in a multiple mirror optical system by inspection.
-notation changes. Most applications of GA have been in theoretical physics, but GA is also a very good +
-analysis tool for engineering. As an example, we use GA to study pattern rotation in optical systems with +
-multiple mirror reflections. The common ways to analyze pattern rotations are to use rotation matrices or +
-optical ray trace codes, but these are often inconvenient. We use GA to develop a simple expression for +
-pattern rotation that is useful for designing or tolerancing pattern rotations in a multiple mirror optical +
-system by inspection.+
  
-  * [[http://www.daehlen.no/adamleon/KA/%5B621%5D%20Object%20Detection%20in%203D%20images%20using%20Conformal%20Geometric.pdf|Object Detection in 3D images using Conformal Geometric Algebra]] (2016) - //Adam Leon Kleppe, Lars Tingelstad, Olav Egeland//+  * [[https://link.springer.com/content/pdf/10.1007%2Fs00006-017-0759-1.pdf|Object Detection in 3D images using Conformal Geometric Algebra]] (2016) - //Adam Leon Kleppe, Lars Tingelstad, Olav Egeland//
 This paper presents an approach for detecting geometric objects in a point cloud from a depth image. The methods in the approach are described and implemented in Conformal Geometric Algebra, resulting in more general, elegant and powerful methods. This paper presents an approach for detecting geometric objects in a point cloud from a depth image. The methods in the approach are described and implemented in Conformal Geometric Algebra, resulting in more general, elegant and powerful methods.
  
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 We present an axiomatic development of geometric algebra. One may think of a geometric algebra as allowing one to add and multiply subspaces of a vector space. Properties of the geometric product are proven and derived products called the wedge and contraction product are introduced. Linear algebraic and geometric concepts such as linear independence and orthogonality may be expressed through the above derived products. Some examples with geometric algebra are then given. We present an axiomatic development of geometric algebra. One may think of a geometric algebra as allowing one to add and multiply subspaces of a vector space. Properties of the geometric product are proven and derived products called the wedge and contraction product are introduced. Linear algebraic and geometric concepts such as linear independence and orthogonality may be expressed through the above derived products. Some examples with geometric algebra are then given.
  
-  * [[http://cronus.uwindsor.ca/users/b/baylis/main.nsf/9d019077a3c4f6768525698a00593654/e639e0cdf0d162c985256bb2004c8fde/$FILE/gaworkbook.pdf|Geometric Algebra: A Unified Mathematical Language +  * [[http://cronus.uwindsor.ca/users/b/baylis/main.nsf/9d019077a3c4f6768525698a00593654/e639e0cdf0d162c985256bb2004c8fde/$FILE/gaworkbook.pdf|Geometric Algebra: A Unified Mathematical Language for Physics]] (2003) - //William Baylis//
-for Physics]] (2003) - //William Baylis//+
 A Workshop at the AAPT Winter Meeting in Austin, Texas, January 12, 2003, organized in cooperation with David Hestenes. A Workshop at the AAPT Winter Meeting in Austin, Texas, January 12, 2003, organized in cooperation with David Hestenes.
  
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 We will do some truly anarchistic computations in basic geometry. We will make these anarchistic computations a part of the establishment. Using the establishment, we will show some quite charming ways of thinking about basic geometry. We will do some truly anarchistic computations in basic geometry. We will make these anarchistic computations a part of the establishment. Using the establishment, we will show some quite charming ways of thinking about basic geometry.
  
-  * [[http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pdf|Computer Graphics using Conformal Geometric Algebra]] (2006) - //Richard James Wareham//+  * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/GA-SIG/Papers/Report.pdf|A Covariant Approach to Geometry using Geometric Algebra]] (2004) - //Anthony Lasenby, Joan Lasenby, Richard Wareham// 
 +This report aims to show that using the mathematical framework of conformal geometric algebra – a 5-dimensional representation of 3-dimensional space – we are able to provide an elegant covariant approach to geometryIn this language, objects such as spheres, circles, lines and planes are simply elements of the algebra and can be transformed and intersected with ease. In addition, rotations, translation, dilations and inversions all become rotations in our 5-dimensional space; we will show how this enables us to provide very simple proofs of complicated constructions. We give examples of the use of this system in computer graphics and indicate how it can be extended into an even more powerful tool – we also discuss its advantages and disadvantages as a programming language. Lastly, we indicate how the framework might possibly be used to unify all geometries, thus enabling us to deal simply with the projective and non-Euclidean cases. 
 + 
 +  * [[https://rjw57.github.io/phd-thesis/rjw-thesis.pdf|Computer Graphics using Conformal Geometric Algebra]] (2006) - //Richard Wareham//
 This thesis investigates the emerging field of Conformal Geometric Algebra (CGA) as a new basis for a CG framework. Computer Graphics is, fundamentally, a particular application of geometry. From a practical standpoint many of the low-level problems to do with rasterising triangles and projecting a three-dimensional world onto a computer screen have been solved and hardware especially designed for this task is available. This thesis investigates the emerging field of Conformal Geometric Algebra (CGA) as a new basis for a CG framework. Computer Graphics is, fundamentally, a particular application of geometry. From a practical standpoint many of the low-level problems to do with rasterising triangles and projecting a three-dimensional world onto a computer screen have been solved and hardware especially designed for this task is available.
  
-  * [[http://home.deib.polimi.it/tubaro/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro//+  * [[https://tubaro.faculty.polimi.it/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro//
 In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible. In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible.
  
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 There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, and quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of the primary goals of 19th century science was to suitably describe vectors in 3-D space. This situation also has the unfortunate consequence of fragmenting knowledge across many disciplines, and requiring a significant amount of time and effort in learning the various formalisms. We thus historically review the development of our various vector systems and conclude that Clifford's multivectors best fulfills the goal of describing vectorial quantities in three dimensions and providing a unified vector system for science. There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, and quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of the primary goals of 19th century science was to suitably describe vectors in 3-D space. This situation also has the unfortunate consequence of fragmenting knowledge across many disciplines, and requiring a significant amount of time and effort in learning the various formalisms. We thus historically review the development of our various vector systems and conclude that Clifford's multivectors best fulfills the goal of describing vectorial quantities in three dimensions and providing a unified vector system for science.
  
-  * [[https://www.researchgate.net/profile/Debashis_Sen/publication/310604392_Representation_of_physical_quantities_From_scalars_vectors_tensors_and_spinors_to_multivectors/links/5833003b08aef19cb81c89e4.pdf|Representation of physical quantities: From scalars, vectors, tensors +  * [[https://www.researchgate.net/profile/Debashis_Sen/publication/310604392_Representation_of_physical_quantities_From_scalars_vectors_tensors_and_spinors_to_multivectors/links/5833003b08aef19cb81c89e4.pdf|Representation of physical quantities: From scalars, vectors, tensors and spinors to multivectors]] (2016) - //D. Sen, Deeprodyuti Sen//
-and spinors to multivectors]] (2016) - //D. Sen, Deeprodyuti Sen//+
 Mathematical representations of physical variables and operators are of primary importance in developing a theory – the relationship among different relevant quantities of any physical process. A thorough account of the representations of different classes of physical variables is drawn up with a brief discussion of various related mathematical systems including quaternion and spinor. The present study would facilitate an introduction to the 'geometric algebra', which provides an immensely productive unification of these systems and promises more. Mathematical representations of physical variables and operators are of primary importance in developing a theory – the relationship among different relevant quantities of any physical process. A thorough account of the representations of different classes of physical variables is drawn up with a brief discussion of various related mathematical systems including quaternion and spinor. The present study would facilitate an introduction to the 'geometric algebra', which provides an immensely productive unification of these systems and promises more.
  
-  * [[https://arxiv.org/pdf/1608.03450v1.pdf|Geometric-Algebra Adaptive Filters]] (2016) - //Wilder B. Lopes, Cassio G. Lopes// +  * [[https://arxiv.org/pdf/1608.03450.pdf|Geometric-Algebra Adaptive Filters]] (2018) - //Wilder B. Lopes, Cassio G. Lopes// 
-This document introduces a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). +This document introduces a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear-algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex-numbers, quaternions etc. Exploiting those characteristics (among others), a general least-squares cost function is posed, from which the GAAFs are designed.
-Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear-algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex-numbers, quaternions etc. Exploiting those characteristics (among others), a general least-squares cost function is posed, from which the GAAFs are designed.+
  
   * [[https://arxiv.org/pdf/1601.06044.pdf|Geometric-Algebra LMS Adaptive Filter and its Application to Rotation Estimation]] (2016) - //Wilder B. Lopes, Anas Al-Nuaimi, Cassio G. Lopes//   * [[https://arxiv.org/pdf/1601.06044.pdf|Geometric-Algebra LMS Adaptive Filter and its Application to Rotation Estimation]] (2016) - //Wilder B. Lopes, Anas Al-Nuaimi, Cassio G. Lopes//
-This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel  GA  least-mean-squares  (GA-LMS)  adaptive  filter,  which inherits properties from standard adaptive filters and from GA, is  developed  to  recursively  estimate   rotor  (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows.  Moreover,  the  employed  GA/GC  framework  allows  for easily applying the resulting filter to estimating rotors in higher dimensions.+This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.
  
-  * [[http://www.lmt.ei.tum.de/forschung/publikationen/dateien/Al-Nuaimi20166DOFPointCloudAlignment.pdf|6DOF Point Cloud Alignment using Geometric Algebra-based Adaptive Filtering]] [[http://wilder.openga.org/wp-content/uploads/2017/03/WACV2016.pdf|(Presentation)]] (2016) - //Anas Al-Nuaimi, Wilder B. Lopes, et al.//+  * [[https://intern.lkn.ei.tum.de/forschung/publikationen/dateien/Al-Nuaimi20166DOFPointCloudAlignment.pdf|6DOF Point Cloud Alignment using Geometric Algebra-based Adaptive Filtering]] [[http://wilder.openga.org/wp-content/uploads/2017/03/WACV2016.pdf|(Presentation)]] (2016) - //Anas Al-Nuaimi, Wilder B. Lopes, et al.//
 In this paper we show that a Geometric Algebra-based least-mean-squares adaptive filter (GA-LMS) can be used to recover the 6-degree-of-freedom alignment of two point clouds related by a set of point correspondences. We present a series of techniques that endow the GA-LMS with outlier (false correspondence) resilience to outperform standard least squares (LS) methods that are based on Singular Value Decomposition (SVD). We furthermore show how to derive and compute the step size of the GA-LMS. In this paper we show that a Geometric Algebra-based least-mean-squares adaptive filter (GA-LMS) can be used to recover the 6-degree-of-freedom alignment of two point clouds related by a set of point correspondences. We present a series of techniques that endow the GA-LMS with outlier (false correspondence) resilience to outperform standard least squares (LS) methods that are based on Singular Value Decomposition (SVD). We furthermore show how to derive and compute the step size of the GA-LMS.
  
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   * [[http://www.naturalspublishing.com/files/published/7c772c2o9nshz1.pdf|Ortogonal Approach for Haptic Rendering Algorithm based in Conformal Geometric Algebra]] (2014) - //Gabriel Sepulveda-Cervantes, Edgar A. Portilla-Flores//   * [[http://www.naturalspublishing.com/files/published/7c772c2o9nshz1.pdf|Ortogonal Approach for Haptic Rendering Algorithm based in Conformal Geometric Algebra]] (2014) - //Gabriel Sepulveda-Cervantes, Edgar A. Portilla-Flores//
-This work presents a novel method for haptic rendering contact force and surface properties for virtual objects using the Conformal Geometric Algebra orthogonal  decomposition approach. The mathematical representation of geometric primitives along with collision algorithms based on its mathematical properties is presented. The orthogonal decomposition of contact and interaction forces is achieved using the same framework and dynamic properties in both subspaces are rendered simultaneously. Comparing with vector calculus,  the Conformal  Geometric Algebra (CGA) approach  provides an easier and more intuitive way to deal with haptic rendering problems due to its inner properties and a simpler representation of geometric objects and linear transformation. The results of the evaluation of the method using a 3 DOF haptic device are presented.+This work presents a novel method for haptic rendering contact force and surface properties for virtual objects using the Conformal Geometric Algebra orthogonal decomposition approach. The mathematical representation of geometric primitives along with collision algorithms based on its mathematical properties is presented. The orthogonal decomposition of contact and interaction forces is achieved using the same framework and dynamic properties in both subspaces are rendered simultaneously. Comparing with vector calculus, the Conformal Geometric Algebra (CGA) approach provides an easier and more intuitive way to deal with haptic rendering problems due to its inner properties and a simpler representation of geometric objects and linear transformation. The results of the evaluation of the method using a 3 DOF haptic device are presented.
  
   * [[https://arxiv.org/pdf/0904.3349v1.pdf|An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota]] (2009) - //Henry Crapo//   * [[https://arxiv.org/pdf/0904.3349v1.pdf|An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota]] (2009) - //Henry Crapo//
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 Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed. Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
  
-  * [[http://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0051756&type=printable|Revisiting Special Relativity: A Natural Algebraic +  * [[http://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0051756&type=printable|Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime]] (2012) - //James M. Chappell, Azhar Iqbal, Nicolangelo Iannella, Derek Abbott//
-Alternative to Minkowski Spacetime]] (2012) - //James M. Chappell, Azhar Iqbal, Nicolangelo Iannella, Derek Abbott//+
 (...) As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary i, with the Clifford bivector i = e_1 e_2 for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis e1 and e2. We find that with this model of planar spacetime, using a two dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane. (...) As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary i, with the Clifford bivector i = e_1 e_2 for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis e1 and e2. We find that with this model of planar spacetime, using a two dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.
  
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 As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems.
  
-  * [[http://journal.frontiersin.org/article/10.3389/fphy.2016.00044/pdf|Time As a Geometric Property of Space]] (2016) - //James M. Chappell, John G. Hartnett, Nicolangelo Iannella, Azhar Iqbal, Derek Abbott// +  * [[https://www.frontiersin.org/articles/10.3389/fphy.2016.00044/full|Time As a Geometric Property of Space]] (2016) - //James M. Chappell, John G. Hartnett, Nicolangelo Iannella, Azhar Iqbal, Derek Abbott// 
-The proper description of time remains a key unsolved problem in science. (...) In order to answer this +The proper description of time remains a key unsolved problem in science. (...) In order to answer this question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension.
-question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension.+
  
   * [[http://aflb.ensmp.fr/AFLB-342/aflb342m679.pdf|The Dirac Equation: an approach through Geometric Algebra]] (2009) - //S. K. Pandey, R. S. Chakravarti//   * [[http://aflb.ensmp.fr/AFLB-342/aflb342m679.pdf|The Dirac Equation: an approach through Geometric Algebra]] (2009) - //S. K. Pandey, R. S. Chakravarti//
 We solve the Dirac equation for a free electron in a manner similar to Toyoki Koga, but using the geometric theory of Clifford algebras which was initiated by David Hestenes. Our solution exhibits a spinning field, among other things. We solve the Dirac equation for a free electron in a manner similar to Toyoki Koga, but using the geometric theory of Clifford algebras which was initiated by David Hestenes. Our solution exhibits a spinning field, among other things.
  
-  * [[https://math.tntech.edu/techreports/TR_2009_3.pdf|Geometric Roots of −1 in Clifford Algebras +  * [[https://math.tntech.edu/techreports/TR_2009_3.pdf|Geometric Roots of −1 in Clifford Algebras Cl(p,q) with p + q ≤ 4]] (2009) - //Eckhard Hitzer, Rafal Ablamowicz// 
-Cl(p,q) with p + q <= 4]] (2009) - //Eckhard Hitzer, Rafal Ablamowicz// +It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of −1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. (...) We now extend this research to general algebras Cl(p,q). We fully derive the geometric roots of −1 for the Clifford (geometric) algebras with p + q ≤ 4.
-It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of −1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. (...) We now extend this research to general algebras Cl(p,q). We fully derive the geometric roots of −1 for the Clifford (geometric) algebras with p + q <= 4.+
  
   * [[http://ijssst.info/Vol-15/No-3/data/3857a357.pdf|Color Image Representation Using Multivector]] (2014) - //Somasis Roy, Anirban Mitra, Sanjit Kumar Setua//   * [[http://ijssst.info/Vol-15/No-3/data/3857a357.pdf|Color Image Representation Using Multivector]] (2014) - //Somasis Roy, Anirban Mitra, Sanjit Kumar Setua//
 This paper discusses about Clifford Algebra and its significance in a color image representation. A color image following RGB color model is described by a multivector in 3D space. The color blades in the multivector define the colors. Different shades of color can be obtained by applying rotor operator on those blades. Clifford color space is introduced to define each color against a pixel of an image in form of a color blade. A multivector function is derived from this space for explaining the color image. In simpler way, it is proposed that a color image is stored in the Clifford color space. This paper is also useful in the reduction of computation time for color image processing as every Grade–k vectors are colors. This paper discusses about Clifford Algebra and its significance in a color image representation. A color image following RGB color model is described by a multivector in 3D space. The color blades in the multivector define the colors. Different shades of color can be obtained by applying rotor operator on those blades. Clifford color space is introduced to define each color against a pixel of an image in form of a color blade. A multivector function is derived from this space for explaining the color image. In simpler way, it is proposed that a color image is stored in the Clifford color space. This paper is also useful in the reduction of computation time for color image processing as every Grade–k vectors are colors.
  
-  * [[http://www.informatik.uni-leipzig.de/~reich/cliffordFT.pdf|Analyzing Real Vector Fields with Clifford +  * [[http://www.informatik.uni-leipzig.de/~reich/cliffordFT.pdf|Analyzing Real Vector Fields with Clifford Convolution and Clifford Fourier Transform]] (2009) - //Wieland Reich, Gerik Scheuermann//
-Convolution and Clifford Fourier Transform]] (2009) - //Wieland Reich, Gerik Scheuermann//+
 Post-processing in computational fluid dynamics as well as processing of fluid flow measurements needs robust methods that can deal with scalar as well as vector fields. While image processing of scalar data is a well-established discipline, there is a lack of similar methods for vector data. This paper surveys a particular approach defining convolution operators on vector fields using geometric algebra. This includes a corresponding Clifford Fourier transform including a convolution theorem. Finally, a comparison is tried with related approaches for a Fourier transform of spatial vector or multivector data. Post-processing in computational fluid dynamics as well as processing of fluid flow measurements needs robust methods that can deal with scalar as well as vector fields. While image processing of scalar data is a well-established discipline, there is a lack of similar methods for vector data. This paper surveys a particular approach defining convolution operators on vector fields using geometric algebra. This includes a corresponding Clifford Fourier transform including a convolution theorem. Finally, a comparison is tried with related approaches for a Fourier transform of spatial vector or multivector data.
  
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 We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wave function. We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wave function.
  
-  * [[http://www.terathon.com/gdc12_lengyel.pdf|Fundamentals of Grassmann Algebra]] (2012) and [[http://www.terathon.com/gdc14_lengyel.pdf|Grassmann Algebra +  * [[http://www.terathon.com/gdc12_lengyel.pdf|Fundamentals of Grassmann Algebra]] (2012) and [[http://www.terathon.com/gdc14_lengyel.pdf|Grassmann Algebra in Game Development]] (2014) - //Eric Lengyel//
-in Game Development]] (2014) - //Eric Lengyel//+
 Presentations at the Game Developers Conference (GDC). Presentations at the Game Developers Conference (GDC).
  
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   * [[http://downloads.hindawi.com/journals/mpe/aip/4687638.pdf|Forward Displacement Analysis of a General 6-3 Stewart Platform Using Conformal Geometric Algebra]] (2016) - //WEI Feng, WEI Shimin, ZHANG Ying, LIAO Qizheng//   * [[http://downloads.hindawi.com/journals/mpe/aip/4687638.pdf|Forward Displacement Analysis of a General 6-3 Stewart Platform Using Conformal Geometric Algebra]] (2016) - //WEI Feng, WEI Shimin, ZHANG Ying, LIAO Qizheng//
 In this paper, a new algorithm for the forward displacement analysis of a general 6-3 Stewart platform (6-3SPS) based on conformal geometric algebra (CGA) is presented. First, a 6-3SPS structure is changed into an equivalent 2RPS-2SPS structure. Then, two kinematic constraint equations are established based on the geometric characteristics, one of which is built according to the point characteristic four-ball intersection in CGA. A 16th-degree univariate polynomial equation is derived from the aforementioned two equations by the Sylvester resultant elimination. Finally, a numerical example is given to verify the algorithm. In this paper, a new algorithm for the forward displacement analysis of a general 6-3 Stewart platform (6-3SPS) based on conformal geometric algebra (CGA) is presented. First, a 6-3SPS structure is changed into an equivalent 2RPS-2SPS structure. Then, two kinematic constraint equations are established based on the geometric characteristics, one of which is built according to the point characteristic four-ball intersection in CGA. A 16th-degree univariate polynomial equation is derived from the aforementioned two equations by the Sylvester resultant elimination. Finally, a numerical example is given to verify the algorithm.
 +
 +  * [[http://downloads.hindawi.com/journals/mpe/aip/6245341.pdf|A Geometric Modeling and Computing Method for Direct Kinematic Analysis of 6-4 Stewart Platforms]] (2018) - //Ying Zhang, Xin Liu et al.//
 +A geometric modeling and solution procedure for direct kinematic analysis of 6-4 Stewart platforms with any link parameters is proposed based on conformal geometric algebra (CGA). Firstly, the positions of the two single spherical joints on the moving platform are formulated by the intersection, dissection, and dual of the basic entities under the frame of CGA. Secondly, a coordinate-invariant equation is derived via CGA operation in the positions of the other two pairwise spherical joints. Thirdly, the other five equations are formulated in terms of geometric constraints. Fourthly, a 32-degree univariate polynomial equation is reduced from a constructed 7 by 7 matrix which is relatively small in size by using a Grobner-Sylvester hybrid method. Finally, a numerical example is employed to verify the solution procedure.
  
   * [[http://www.ias.ac.in/article/fulltext/reso/021/12/1105-1117|Geometry of Spin: Clifford Algebraic Approach]] (2016) - //Rukhsan-Ul-Haq//   * [[http://www.ias.ac.in/article/fulltext/reso/021/12/1105-1117|Geometry of Spin: Clifford Algebraic Approach]] (2016) - //Rukhsan-Ul-Haq//
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 Due to the variable curvature of the conformal carrier, the pattern of each element has a different direction. The traditional method of analyzing the conformal array is to use the Euler rotation angle and its matrix representation. However, it is computationally demanding especially for irregular array structures. In this paper, we present a novel algorithm by combining the geometric algebra with Multiple Signal Classification (MUSIC), termed as GA-MUSIC, to solve the direction of arrival (DOA) for cylindrical conformal array. And on this basis, we derive the pattern and array manifold. Compared with the existing algorithms, our proposed one avoids the cumbersome matrix transformations and largely decreases the computational complexity. The simulation results verify the effectiveness of the proposed method. Due to the variable curvature of the conformal carrier, the pattern of each element has a different direction. The traditional method of analyzing the conformal array is to use the Euler rotation angle and its matrix representation. However, it is computationally demanding especially for irregular array structures. In this paper, we present a novel algorithm by combining the geometric algebra with Multiple Signal Classification (MUSIC), termed as GA-MUSIC, to solve the direction of arrival (DOA) for cylindrical conformal array. And on this basis, we derive the pattern and array manifold. Compared with the existing algorithms, our proposed one avoids the cumbersome matrix transformations and largely decreases the computational complexity. The simulation results verify the effectiveness of the proposed method.
  
-  * [[https://www.researchgate.net/profile/Ovidiu_Cristinel_Stoica/publication/308062409_The_Standard_Model_Algebra/links/57d8746308ae601b39afab89.pdf|The Standard Model Algebra]] (2016) - //Ovidiu Cristinel Stoica// +  * [[https://arxiv.org/pdf/1702.04336v3.pdf|The Standard Model Algebra]] (2018) - //Ovidiu Cristinel Stoica// 
-A simple algebra is shown to contain automatically the leptons and quarks of a generic generation of the Standard Model, and the electromagnetic and color gauge symmetries. The algebra is just the Clifford algebra of a complex six-dimensional vector space endowed with preferred Witt decomposition. Its ideals determined by the Witt decomposition correspond naturally to leptons and quarks. It is shown that the algebra is invariant to the electromagnetic and color symmetrieswhich are generated by its spinors. The weak symmetry is also present already in its broken form, but this part is still under development. The Dirac algebra is obtained as a distinguished subalgebra acting on the ideals representing leptons and quarks.+A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra Cl6, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify themresulting in the model proposed here.
  
   * [[https://arxiv.org/pdf/1611.09182.pdf|Standard model physics from an algebra?]] (2015) - //Cohl Furey//   * [[https://arxiv.org/pdf/1611.09182.pdf|Standard model physics from an algebra?]] (2015) - //Cohl Furey//
-This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra R⊗C⊗H⊗O, the tensor +This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra R⊗C⊗H⊗O, the tensor product of the only four normed division algebras over the real numbers. Using nothing more than R⊗C⊗H⊗O acting on itself, we set out to find standard model particle representations: a task which occupies the remainder of this text.
-product of the only four normed division algebras over the real numbers. Using nothing more than R⊗C⊗H⊗O acting  on  itself, we set out to find standard model particle representations: a task which occupies the remainder of this text.+
  
   * [[http://www.mdpi.com/2073-8994/8/9/92/pdf|Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra]] (2016) - //Juan C. Bravo, Manuel V. Castilla//   * [[http://www.mdpi.com/2073-8994/8/9/92/pdf|Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra]] (2016) - //Juan C. Bravo, Manuel V. Castilla//
 (...) a new circuit analysis approach using geometric algebra is used to develop the most general proof of energy conservation in industrial building loads. In terms of geometric objects, this powerful tool calculates the voltage, current, and apparent power in electrical systems in non-sinusoidal, linear/nonlinear situations. In contrast to the traditional method developed by Steinmetz, the suggested powerful tool extends the concept of phasor to multivector-phasors and is performed in a new Generalized Complex Geometric Algebra structure (CGn), where Gn is the Clifford algebra in n-dimensional real space and C is the complex vector space. (...) a new circuit analysis approach using geometric algebra is used to develop the most general proof of energy conservation in industrial building loads. In terms of geometric objects, this powerful tool calculates the voltage, current, and apparent power in electrical systems in non-sinusoidal, linear/nonlinear situations. In contrast to the traditional method developed by Steinmetz, the suggested powerful tool extends the concept of phasor to multivector-phasors and is performed in a new Generalized Complex Geometric Algebra structure (CGn), where Gn is the Clifford algebra in n-dimensional real space and C is the complex vector space.
  
-  * [[http://phydid.physik.fu-berlin.de/index.php/phydid-b/article/download/642/769| +  * [[http://phydid.physik.fu-berlin.de/index.php/phydid-b/article/download/642/769|Sandwich Products and Reflections]] (2015) - //Martin Erik Horn//
-Sandwich Products and Reflections]] (2015) - //Martin Erik Horn//+
 As reflections are an elementary part of model construction in physics, we really should look for a mathematical picture which allows for a very general description of reflections. The sandwich product delivers such a picture. Using the mathematical language of Geometric Algebra, reflections at vectors of arbitrary dimensions and reflections at multivectors (i.e. at linear combinations of vectors of arbitrary dimensions) can be described mathematically in an astonishingly coherent picture. As reflections are an elementary part of model construction in physics, we really should look for a mathematical picture which allows for a very general description of reflections. The sandwich product delivers such a picture. Using the mathematical language of Geometric Algebra, reflections at vectors of arbitrary dimensions and reflections at multivectors (i.e. at linear combinations of vectors of arbitrary dimensions) can be described mathematically in an astonishingly coherent picture.
  
-  * [[http://repo.flib.u-fukui.ac.jp/dspace/bitstream/10098/3298/1/AN00215401-049-02-016.pdf|Antisymmetric Matrices Are Real Bivectors]] (2001) - //Eckhard Hitzer//+  * [[http://vixra.org/pdf/1306.0112v1.pdf|Antisymmetric Matrices Are Real Bivectors]] (2001) - //Eckhard Hitzer//
 This paper briefly reviews the conventional method of obtaining the canonical form of an antisymmetric (skewsymmetric, alternating) matrix. Conventionally a vector space over the complex field has to be introduced. After a short introduction to the universal mathematical "language" Geometric Calculus, its fundamentals, i.e. its "grammar" Geometric Algebra (Clifford Algebra) is explained. This lays the groundwork for its real geometric and coordinate free application in order to obtain the canonical form of an antisymmetric matrix in terms of a bivector, which is isomorphic to the conventional canonical form. Then concrete applications to two, three and four dimensional antisymmetric square matrices follow. Because of the physical importance of the Minkowski metric, the canonical form of an antisymmetric matrix with respect to the Minkowski metric is derived as well. A final application to electromagnetic fields concludes the work. This paper briefly reviews the conventional method of obtaining the canonical form of an antisymmetric (skewsymmetric, alternating) matrix. Conventionally a vector space over the complex field has to be introduced. After a short introduction to the universal mathematical "language" Geometric Calculus, its fundamentals, i.e. its "grammar" Geometric Algebra (Clifford Algebra) is explained. This lays the groundwork for its real geometric and coordinate free application in order to obtain the canonical form of an antisymmetric matrix in terms of a bivector, which is isomorphic to the conventional canonical form. Then concrete applications to two, three and four dimensional antisymmetric square matrices follow. Because of the physical importance of the Minkowski metric, the canonical form of an antisymmetric matrix with respect to the Minkowski metric is derived as well. A final application to electromagnetic fields concludes the work.
  
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 A wave equation with mass term is studied for all fermionic particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks u and d with three states of color and antiquarks u' and d'. This wave equation is form invariant under the Cl*_3 group generalizing the relativistic invariance. It is gauge invariant under the U(1)*SU(2)*SU(3) group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra Cl_1,5. Then many features of the standard model, charge conjugation, color, left waves, and Lagrangian formalism, are obtained in the frame of the first quantization. A wave equation with mass term is studied for all fermionic particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks u and d with three states of color and antiquarks u' and d'. This wave equation is form invariant under the Cl*_3 group generalizing the relativistic invariance. It is gauge invariant under the U(1)*SU(2)*SU(3) group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra Cl_1,5. Then many features of the standard model, charge conjugation, color, left waves, and Lagrangian formalism, are obtained in the frame of the first quantization.
  
-  * [[http://file.scirp.org/pdf/JMP_2016052713551874.pdf|Three Clifford Algebras for Four Kinds of +  * [[http://file.scirp.org/pdf/JMP_2016052713551874.pdf|Three Clifford Algebras for Four Kinds of Interactions]] (2016) - //Claude Daviau, Jacques Bertrand//
-Interactions]] (2016) - //Claude Daviau, Jacques Bertrand//+
 Three Clifford algebras are sufficient to describe all interactions of modern physics: The Clifford algebra of the usual space is enough to describe all aspects of electromagnetism, including the quantum wave of the electron. The Clifford algebra of space-time is enough for electro-weak interactions. To get the gauge group of the standard model, with electro-weak and strong interactions, a third algebra is sufficient, with only two more dimensions of space. The Clifford algebra of space allows us to include also gravitation. We discuss the advantages of our approach. Three Clifford algebras are sufficient to describe all interactions of modern physics: The Clifford algebra of the usual space is enough to describe all aspects of electromagnetism, including the quantum wave of the electron. The Clifford algebra of space-time is enough for electro-weak interactions. To get the gauge group of the standard model, with electro-weak and strong interactions, a third algebra is sufficient, with only two more dimensions of space. The Clifford algebra of space allows us to include also gravitation. We discuss the advantages of our approach.
  
-  * [[http://file.scirp.org/pdf/JMP_2016083114210126.pdf|Towards the Unification of All Interactions +  * [[http://file.scirp.org/pdf/JMP_2016083114210126.pdf|Towards the Unification of All Interactions (The First Part: The Spinor Wave)]] (2016) - //Claude Daviau, Jacques Bertrand, Dominique Girardot//
-(The First Part: The Spinor Wave)]] (2016) - //Claude Daviau, Jacques Bertrand, Dominique Girardot//+
 For the unification of gravitation with electromagnetism, weak and strong interactions, we use a unique and very simple framework, the Clifford algebra of space Cl_3 = M_2(C). We enlarge our previous wave equation to the general case, including all leptons, quarks and antiparticles of the first generation. The wave equation is a generalization of the Dirac equation with a compulsory non-linear mass term. This equation is form invariant under the Cl_3^∗ group of the invertible elements in the space algebra. The form invariance is fully compatible with the U(1)*SU(2)*SU(3) gauge invariance of the standard model. For the unification of gravitation with electromagnetism, weak and strong interactions, we use a unique and very simple framework, the Clifford algebra of space Cl_3 = M_2(C). We enlarge our previous wave equation to the general case, including all leptons, quarks and antiparticles of the first generation. The wave equation is a generalization of the Dirac equation with a compulsory non-linear mass term. This equation is form invariant under the Cl_3^∗ group of the invertible elements in the space algebra. The form invariance is fully compatible with the U(1)*SU(2)*SU(3) gauge invariance of the standard model.
  
-  * [[http://file.scirp.org/pdf/JMP_2016123011142663.pdf|Towards the Unification, Part 2: Simplified +  * [[http://file.scirp.org/pdf/JMP_2016123011142663.pdf|Towards the Unification, Part 2: Simplified Equations, Covariant Derivative, Photons]] (2016) - //Claude Daviau, Jacques Bertrand, Dominique Girardot//
-Equations, Covariant Derivative, Photons]] (2016) - //Claude Daviau, Jacques Bertrand, Dominique Girardot//+
 We continue the study of the Standard Model of Quantum Physics in the Clifford algebra of space. We get simplified mass terms for the fermion part of the wave. We insert the simplified equations in the frame of General Relativity. We construct the electromagnetic field of the photon, alone boson without proper mass. We explain how the Pauli principle comes from the equivalence principle of General Relativity. We transpose in the frame of the algebra of space the second quantification of the electromagnetic field. We discuss the changes introduced here. We continue the study of the Standard Model of Quantum Physics in the Clifford algebra of space. We get simplified mass terms for the fermion part of the wave. We insert the simplified equations in the frame of General Relativity. We construct the electromagnetic field of the photon, alone boson without proper mass. We explain how the Pauli principle comes from the equivalence principle of General Relativity. We transpose in the frame of the algebra of space the second quantification of the electromagnetic field. We discuss the changes introduced here.
  
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 A primitive shape detection algorithm is implemented in a C++ based software. The algorithm is implemented for planes, spheres and cylinders. Results show that the algorithm is able to detect the shapes in data sets containing up to 90% outliers. Furthermore, a real-time tracking algorithm based on the primitive shape detection algorithm is implemented to track primitives in a real-time data stream from a 3D camera. The run-time of the tracking algorithm is well below the required rate for a 60 frames per second data stream. A multiple shape detection algorithm is also developed. The goal is to detect multiple shapes in a point cloud with a single run of the algorithm. The algorithm is implemented for spheres and results show that multiple spheres can be successfully detected in a point cloud. The accuracy and efficiency of the algorithms is demonstrated in a robotic pick-and-place task. A primitive shape detection algorithm is implemented in a C++ based software. The algorithm is implemented for planes, spheres and cylinders. Results show that the algorithm is able to detect the shapes in data sets containing up to 90% outliers. Furthermore, a real-time tracking algorithm based on the primitive shape detection algorithm is implemented to track primitives in a real-time data stream from a 3D camera. The run-time of the tracking algorithm is well below the required rate for a 60 frames per second data stream. A multiple shape detection algorithm is also developed. The goal is to detect multiple shapes in a point cloud with a single run of the algorithm. The algorithm is implemented for spheres and results show that multiple spheres can be successfully detected in a point cloud. The accuracy and efficiency of the algorithms is demonstrated in a robotic pick-and-place task.
  
-  * [[http://gacomputing.info/wp-content/uploads/2016/07/ga-computing-2016-v1.pdf|Practical Computing with Geometric Algebra Converting Basic Geometric Algebra Relations to Computations on Multivector Coordinates]] (2016) - //Ahmad Hosny Eid//+  * [[https://gmac-guides.netlify.com/wp-content/uploads/2016/07/ga-computing-2016-v1.pdf|Practical Computing with Geometric Algebra Converting Basic Geometric Algebra Relations to Computations on Multivector Coordinates]] (2016) - //Ahmad Hosny Eid//
 This article provides a summary, without proofs, of the fundamental algebraic concepts and operations of GA. After this, the article contains an explanation of how to transform high-level mathematical GA products and algebraic operations into equivalent lower-level computations on multivector coordinates. The aim is to provide a computational basis for implementing compilers that can automatically perform such conversion for the purpose of efficient software implementations of GA-based models and algorithms. This article provides a summary, without proofs, of the fundamental algebraic concepts and operations of GA. After this, the article contains an explanation of how to transform high-level mathematical GA products and algebraic operations into equivalent lower-level computations on multivector coordinates. The aim is to provide a computational basis for implementing compilers that can automatically perform such conversion for the purpose of efficient software implementations of GA-based models and algorithms.
  
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   * [[https://www.researchgate.net/publication/264423339_An_invitation_to_Clifford_Analysis|Una Invitación al Análisis de Clifford]] (2003) - //Richard Delanghe, Juan Bory-Reyes//   * [[https://www.researchgate.net/publication/264423339_An_invitation_to_Clifford_Analysis|Una Invitación al Análisis de Clifford]] (2003) - //Richard Delanghe, Juan Bory-Reyes//
-Una panorámica  de los tópicos principales y herramientas básicas del Análisis de  Clifford se presenta en este artículo, al mismo tiempo, las principales fórmulas integrales relacionadas con la integral tipo Cauchy --- y  su versión  singular --- son  analizadas en un contexto multidimensional, con el uso de las  técnicas de álgebras de  Clifford. Se incluyen también algunas notas históricas sobre el desarrollo de este campo de investigación.+Una panorámica de los tópicos principales y herramientas básicas del Análisis de Clifford se presenta en este artículo, al mismo tiempo, las principales fórmulas integrales relacionadas con la integral tipo Cauchy --- y su versión singular --- son analizadas en un contexto multidimensional, con el uso de las técnicas de álgebras de Clifford. Se incluyen también algunas notas históricas sobre el desarrollo de este campo de investigación.
  
   * [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano//   * [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano//
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   * [[http://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IFAC_ICINCO_2007/ICINCO%202007/Area%202%20-%20Robotics%20and%20Automation/Volume%202/Short%20Papers/C2_496_Bayro-Corrochano.pdf|Geometric Advanced Techniques for Robot Grasping using Stereoscopic Vision]] (2007) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//   * [[http://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IFAC_ICINCO_2007/ICINCO%202007/Area%202%20-%20Robotics%20and%20Automation/Volume%202/Short%20Papers/C2_496_Bayro-Corrochano.pdf|Geometric Advanced Techniques for Robot Grasping using Stereoscopic Vision]] (2007) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//
-In this paper the authors propose geometric techniques to deal with the problem of grasping objects relying on their mathematical models.  For that we use the geometric algebra framework to formulate the kinematics of a three finger robotic hand. Our main objective is by formulating a kinematic control law to close the loop between perception and actions. This allows us to perform a smooth visually guided object grasping action.+In this paper the authors propose geometric techniques to deal with the problem of grasping objects relying on their mathematical models. For that we use the geometric algebra framework to formulate the kinematics of a three finger robotic hand. Our main objective is by formulating a kinematic control law to close the loop between perception and actions. This allows us to perform a smooth visually guided object grasping action.
  
   * [[http://downloads.hindawi.com/journals/abb/2011/728132.pdf|Robot Object Manipulation Using Stereoscopic Vision and Conformal Geometric Algebra]] (2011) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//   * [[http://downloads.hindawi.com/journals/abb/2011/728132.pdf|Robot Object Manipulation Using Stereoscopic Vision and Conformal Geometric Algebra]] (2011) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//
 This paper uses geometric algebra to formulate, in a single framework, the kinematics of a three finger robotic hand, a binocular robotic head, and the interactions between 3D objects, all of which are seen in stereo images. The main objective is the formulation of a kinematic control law to close the loop between perception and actions, which allows to perform a smooth visually guided object manipulation. This paper uses geometric algebra to formulate, in a single framework, the kinematics of a three finger robotic hand, a binocular robotic head, and the interactions between 3D objects, all of which are seen in stereo images. The main objective is the formulation of a kinematic control law to close the loop between perception and actions, which allows to perform a smooth visually guided object manipulation.
- 
  
   * [[http://downloads.hindawi.com/journals/tswj/2013/463983.pdf|Control of Disturbing Loads in Residential and Commercial Buildings via Geometric Algebra]] (2013) - //Manuel-V. Castilla//   * [[http://downloads.hindawi.com/journals/tswj/2013/463983.pdf|Control of Disturbing Loads in Residential and Commercial Buildings via Geometric Algebra]] (2013) - //Manuel-V. Castilla//
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 We study "constrained generalized Killing (s)pinors," which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. We study "constrained generalized Killing (s)pinors," which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems.
  
-   * [[http://downloads.hindawi.com/journals/mpe/2016/7412373.pdf|Unified Spatial Intersection Algorithms Based on Conformal Geometric Algebra]] (2016) - //Feng Zhang et al.// +  * [[http://downloads.hindawi.com/journals/mpe/2016/7412373.pdf|Unified Spatial Intersection Algorithms Based on Conformal Geometric Algebra]] (2016) - //Feng Zhang et al.// 
- In this study, we propose algorithms based on Conformal Geometric Algebra to determine the spatial relationships between geographic objects in a unified manner. The unified representation and intersection computation can be realized for geometric objects of different dimensions. Different basic judgment rules are provided for different simple geometries. The algorithms are designed and implemented using MapReduce to improve the efficiency of the algorithms.+In this study, we propose algorithms based on Conformal Geometric Algebra to determine the spatial relationships between geographic objects in a unified manner. The unified representation and intersection computation can be realized for geometric objects of different dimensions. Different basic judgment rules are provided for different simple geometries. The algorithms are designed and implemented using MapReduce to improve the efficiency of the algorithms.
  
   * [[http://downloads.hindawi.com/journals/ijmms/2002/427561.pdf|Szegő projections for hardy spaces of monogenic functions and applications]] (2002) - //Swanhild Bernstein, Loredana Lanzani//   * [[http://downloads.hindawi.com/journals/ijmms/2002/427561.pdf|Szegő projections for hardy spaces of monogenic functions and applications]] (2002) - //Swanhild Bernstein, Loredana Lanzani//
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   * [[http://cs.unitbv.ro/~acami/what_kc_can_do.pdf|What the Kähler Calculus can do that other calculi cannot]] (2016) - //Jose G. Vargas//   * [[http://cs.unitbv.ro/~acami/what_kc_can_do.pdf|What the Kähler Calculus can do that other calculi cannot]] (2016) - //Jose G. Vargas//
 Underlied by Clifford algebra of differential forms --- like tangent Clifford algebra underlies the geometric calculus --- it brings about a fresh new view of quantum mechanics. This view arises, almost without effort, from the equation which is in Kähler Calculus what the Dirac equation is in traditional quantum mechanics. Underlied by Clifford algebra of differential forms --- like tangent Clifford algebra underlies the geometric calculus --- it brings about a fresh new view of quantum mechanics. This view arises, almost without effort, from the equation which is in Kähler Calculus what the Dirac equation is in traditional quantum mechanics.
 +
 +  * [[https://www.mit.edu/~fengt/282C.pdf|The Atiyah–Singer index theorem]] (2015) - //Dan Berwick-Evans, via Tony Feng//
 +Lecture notes about the Atiyah-Singer index theorem.
 +
 +  * [[http://www.cs.ox.ac.uk/people/david.reutter/AtiyahSinger_Essay.pdf|The Heat Equation and the Atiyah-Singer Index Theorem]] (2015) - //David Reutter//
 +The Atiyah-Singer index theorem is a milestone of twentieth century mathematics. Roughly speaking, it relates a global analytical datum of a manifold --- the number of solutions of a certain linear PDE --- to an integral of local topological expressions over this manifold. The index theorem provided a link between analysis, geometry and topology, paving the way for many further applications along these lines.
  
   * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie//   * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie//
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 Even though it has been almost a century since quantum mechanics planted roots, the field has its share of unresolved problems. It could be the result of a wrong mathematical structure providing inadequate understanding of the quantum phenomena.  Even though it has been almost a century since quantum mechanics planted roots, the field has its share of unresolved problems. It could be the result of a wrong mathematical structure providing inadequate understanding of the quantum phenomena. 
  
-  * [[http://www.i-eos.org:8080/ieos/OpenXchive/adg2016/adg2016-paper/at_download/file|A New Formalization of Origami in Geometric Algebra]] (2016) - //Tetsuo Ida, Jacques Fleuriot, and Fadoua Ghourabi//+  * [[https://hal.inria.fr/hal-01334334/document|A New Formalization of Origami in Geometric Algebra]] (2016) - //Tetsuo Ida, Jacques Fleuriot, and Fadoua Ghourabi//
 We present a new formalization of origami modeling and theorem proving using a geometric algebra. We formalize in Isabelle/HOL a geometric algebra G_3 to treat origamis in both 2D and 3D physical space. We define G_3 as a type class of Isabelle/HOL. The objects in G_3 are multivectors. We prove that the co-datatype of a multivector is an element instance of the type class G_3. We prove by Isabelle/HOL a large number of identities and equations that hold in G_3. With G_3 we then reformulate Huzita’s elementary origami folds in equations of multivectors. We present a new formalization of origami modeling and theorem proving using a geometric algebra. We formalize in Isabelle/HOL a geometric algebra G_3 to treat origamis in both 2D and 3D physical space. We define G_3 as a type class of Isabelle/HOL. The objects in G_3 are multivectors. We prove that the co-datatype of a multivector is an element instance of the type class G_3. We prove by Isabelle/HOL a large number of identities and equations that hold in G_3. With G_3 we then reformulate Huzita’s elementary origami folds in equations of multivectors.
  
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 This paper provides an implementation of a novel signal processing co-processor using a Geometric Algebra technique tailored for fast and complex geometric calculations in multiple dimensions. This is the first hardware implementation of Geometric Algebra to specifically address the issue of scalability to multiple (1-8) dimensions. This paper presents a detailed description of the implementation, with a particular focus on the techniques of optimization used to improve performance. Results are presented which demonstrate at least 3x performance improvements compared to previously published work. This paper provides an implementation of a novel signal processing co-processor using a Geometric Algebra technique tailored for fast and complex geometric calculations in multiple dimensions. This is the first hardware implementation of Geometric Algebra to specifically address the issue of scalability to multiple (1-8) dimensions. This paper presents a detailed description of the implementation, with a particular focus on the techniques of optimization used to improve performance. Results are presented which demonstrate at least 3x performance improvements compared to previously published work.
  
-  * [[http://www.springer.com/cda/content/document/cda_downloaddocument/9783319306735-c2.pdf| +  * [[http://www.springer.com/cda/content/document/cda_downloaddocument/9783319306735-c2.pdf|Singularity Analysis of a Novel Minimally-Invasive-Surgery Hybrid Robot Using Geometric Algebra]] (2016) - //Tanio K. Kanev//
-Singularity Analysis of a Novel Minimally-Invasive-Surgery Hybrid Robot Using Geometric Algebra]] (2016) - //Tanio K. Kanev//+
 The paper presents an analysis of the singularities of a novel type of medical robot for minimally invasive surgery (MIS) using the language of the geometric algebra. The analysis focuses on the parallel manipulator, which is the key component of the robot. The proposed new parallel manipulator provides a remote centre of motion located at the incision point of the patient’s body. The aim of the paper is to derive the geometric condition for singularity in terms of geometric algebra and thus to reveal the singular configurations in order to avoid them during the surgical procedure. The obtained geometric condition for singularity leads further to the derivation of the algebraic formulation of the singularity surface which is graphically presented.  The paper presents an analysis of the singularities of a novel type of medical robot for minimally invasive surgery (MIS) using the language of the geometric algebra. The analysis focuses on the parallel manipulator, which is the key component of the robot. The proposed new parallel manipulator provides a remote centre of motion located at the incision point of the patient’s body. The aim of the paper is to derive the geometric condition for singularity in terms of geometric algebra and thus to reveal the singular configurations in order to avoid them during the surgical procedure. The obtained geometric condition for singularity leads further to the derivation of the algebraic formulation of the singularity surface which is graphically presented. 
  
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   * [[http://afrodita.zcu.cz/~skala/PUBL/PUBL_2017/2017-CoMeSySo-Laplace-GA-FINAL.pdf|Geometric Algebra, Extended Cross-Product and Laplace Transform for Multidimensional Dynamical Systems]] (2017) - //Vaclav Skala//   * [[http://afrodita.zcu.cz/~skala/PUBL/PUBL_2017/2017-CoMeSySo-Laplace-GA-FINAL.pdf|Geometric Algebra, Extended Cross-Product and Laplace Transform for Multidimensional Dynamical Systems]] (2017) - //Vaclav Skala//
-This contribution describes a new approach for solving linear system of algebraic equations and differential equations using Laplace transform by the extended-cross product. It will be shown that a solution of a linear system of equations Ax=0 or Ax=b is equivalent to the extended cross-product if the projective extension of the Euclidean system and the principle of duality are used. Using the Laplace transform differential equations are transformed to a system of linear algebraic equations, which can be solved using the extended cross-product (outer product). The presented approach enables to avoid division operation and extents numerical precision as well. It also offers applications of matrix-vector and vector-vector operations in symbolic manipulation, which can leads to new algorithms and/or new formula. The proposed approach can be applied also for stability evaluation of dynamical systems. In the case of numerical computation, it supports vector operation and SSE instructions or GPU can be used efficiently. +This contribution describes a new approach for solving linear system of algebraic equations and differential equations using Laplace transform by the extended-cross product. It will be shown that a solution of a linear system of equations Ax=0 or Ax=b is equivalent to the extended cross-product if the projective extension of the Euclidean system and the principle of duality are used. Using the Laplace transform differential equations are transformed to a system of linear algebraic equations, which can be solved using the extended cross-product (outer product). The presented approach enables to avoid division operation and extents numerical precision as well. It also offers applications of matrix-vector and vector-vector operations in symbolic manipulation, which can leads to new algorithms and/or new formula. The proposed approach can be applied also for stability evaluation of dynamical systems. In the case of numerical computation, it supports vector operation and SSE instructions or GPU can be used efficiently
 + 
 +  * [[http://afrodita.zcu.cz/~skala/PUBL/PUBL_2018/2018-EECS-Laplace-Bern.pdf|Geometric Product for Multidimensional Dynamical Systems - Laplace Transform and Geometric Algebra]] (2018) - //Vaclav Skala, Michal Smolik, Mariia Martynova// 
 +This contribution describes a new approach to a solution of multidimensional dynamical systems using the Laplace transform and geometrical product, i.e. using inner product (dot product, scalar product) and outer product (extended cross-product). It leads to a linear system of equations Ax=0 or Ax=b which is equivalent to the outer product if the projective extension of the Euclidean system and the principle of duality are used. The paper explores property of the geometrical product in the frame of multidimensional dynamical system.
  
   * [[https://www.researchgate.net/publication/318929234_Type_Synthesis_of_Parallel_Tracking_Mechanism_with_Varied_Axes_by_Modeling_Its_Finite_Motions_Algebraically|Type Synthesis of Parallel Tracking Mechanism with Varied Axes by Modeling Its Finite Motions Algebraically]] (2017) - //Yang Qi, Tao Sun, Yimin Song//   * [[https://www.researchgate.net/publication/318929234_Type_Synthesis_of_Parallel_Tracking_Mechanism_with_Varied_Axes_by_Modeling_Its_Finite_Motions_Algebraically|Type Synthesis of Parallel Tracking Mechanism with Varied Axes by Modeling Its Finite Motions Algebraically]] (2017) - //Yang Qi, Tao Sun, Yimin Song//
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   * [[https://arxiv.org/pdf/1705.06668.pdf|Introducing Geometric Algebra to Geometric Computing Software Developers: A Computational Thinking Approach]] (2017) - //Ahmad Hosny Eid//   * [[https://arxiv.org/pdf/1705.06668.pdf|Introducing Geometric Algebra to Geometric Computing Software Developers: A Computational Thinking Approach]] (2017) - //Ahmad Hosny Eid//
 Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying algebraic system to describe geometric models, the use of software abstractions alone can result in many design and maintenance problems. Geometric Algebra (GA) can be a universal abstract algebraic language for software engineering geometric computing applications. Few sources, however, provide enough information about GA-based software implementations targeting the software engineering community. In particular, successfully introducing GA to software engineers requires quite different approaches from introducing GA to mathematicians or physicists. This article provides a high-level introduction to the abstract concepts and algebraic representations behind the elegant GA mathematical structure. The article focuses on the conceptual and representational abstraction levels behind GA mathematics with sufficient references for more details. In addition, the article strongly recommends applying the methods of Computational Thinking in both introducing GA to software engineers, and in using GA as a mathematical language for developing Geometric Computing software systems. Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying algebraic system to describe geometric models, the use of software abstractions alone can result in many design and maintenance problems. Geometric Algebra (GA) can be a universal abstract algebraic language for software engineering geometric computing applications. Few sources, however, provide enough information about GA-based software implementations targeting the software engineering community. In particular, successfully introducing GA to software engineers requires quite different approaches from introducing GA to mathematicians or physicists. This article provides a high-level introduction to the abstract concepts and algebraic representations behind the elegant GA mathematical structure. The article focuses on the conceptual and representational abstraction levels behind GA mathematics with sufficient references for more details. In addition, the article strongly recommends applying the methods of Computational Thinking in both introducing GA to software engineers, and in using GA as a mathematical language for developing Geometric Computing software systems.
 +
 +  * [[https://core.ac.uk/download/pdf/153543582.pdf|Generalized Bernoulli Numbers and Polynomials in the Context of the Clifford Analysis]] (2017) - //Sreelatha Chandragiri, Olga A. Shishkina//
 +In this paper, we consider the generalization of the Bernoulli numbers and polynomials for the case of the hypercomplex variables. Multidimensional analogs of the main properties of classic polynomials are proved.
  
   * [[https://arxiv.org/pdf/1711.02641.pdf|Heisenberg's and Hardy's Uncertainty Principles in Real Clifford Algebras]] (2017) - //Rim Jday//   * [[https://arxiv.org/pdf/1711.02641.pdf|Heisenberg's and Hardy's Uncertainty Principles in Real Clifford Algebras]] (2017) - //Rim Jday//
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   * [[https://arxiv.org/pdf/1711.02513.pdf|CGAlgebra: a Mathematica package for conformal geometric algebra]] (2017) - //Jose L. Aragon//   * [[https://arxiv.org/pdf/1711.02513.pdf|CGAlgebra: a Mathematica package for conformal geometric algebra]] (2017) - //Jose L. Aragon//
 A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the calculations of geometric, outer and inner products, as well as many other calculations related with geometric transformations. A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the calculations of geometric, outer and inner products, as well as many other calculations related with geometric transformations.
 +
 +  * [[https://hal.archives-ouvertes.fr/hal-01622682/document|Differential Geometry Revisited by Biquaternion Clifford Algebra]] (2017) - //Patrick Girard et al.//
 +In the last century, differential geometry has been expressed within various calculi: vectors, tensors, spinors, exterior differential forms and recently Clifford algebras. Clifford algebras yield an excellent representation of the rotation group and of the Lorentz group which are the cornerstones of the theory of moving frames. Though Clifford algebras are all related to quaternions via the Clifford theorem, a biquaternion formulation of differential geometry does not seem to have been formulated so far. The paper develops, in 3D Euclidean space, a biquaternion calculus, having an associative exterior product, and applies it to differential geometry. The formalism being new, the approach is intended to be pedagogical. Since the methods of Clifford algebras are similar in other dimensions, it is hoped that the paper might open new perspectives for a 4D hyperbolic differential geometry. All the calculi presented here can easily be implemented algebraically on Mathematica and numerically on Matlab. Examples, matrix representations, and a Mathematica worksheet are provided.
 +
 +  * [[https://hal.archives-ouvertes.fr/hal-01572459/document|Free-form additive motions using conformal geometric algebra]] (2017) - //Mat Hunt, Glen Mullineux, Robert J Cripps, Ben Cross//
 +Free-form motions in B-spline form can be created from a number of prescribed control poses using the de Casteljau algorithm. With poses defined using conformal geometric algebra, it is natural to combine poses multiplicatively. Additive combinations offer alternative freedoms in design and avoid dealing with noninteger exponents. This paper investigates additive combinations and shows how to modify the conventional conformal geometric algebra definitions to allow such combinations to be well-defined. The additive and multiplicative approaches are compared and in general they generate similar motions, with the additive approach offering computational simplicity.
  
   * [[https://www.researchgate.net/profile/Giorgio_Vassallo/publication/320274378_Maxwell%27s_Equations_and_Occam%27s_Razor/links/59da8fa8a6fdcc2aad12a733/Maxwells-Equations-and-Occams-Razor.pdf|Maxwell’s Equations and Occam’s Razor]] (2017) - //Francesco Celani, Antonino Oscar Di Tommaso, Giorgio Vassallo//   * [[https://www.researchgate.net/profile/Giorgio_Vassallo/publication/320274378_Maxwell%27s_Equations_and_Occam%27s_Razor/links/59da8fa8a6fdcc2aad12a733/Maxwells-Equations-and-Occams-Razor.pdf|Maxwell’s Equations and Occam’s Razor]] (2017) - //Francesco Celani, Antonino Oscar Di Tommaso, Giorgio Vassallo//
 A straightforward application of Occam’s razor principle to Maxwell’s equation shows that only one entity, the electromagnetic four-potential, is at the origin of a plurality of concepts and entities in physics. The application of the so called "Lorenz gauge" in Maxwell's equations denies the status of real physical entity to a scalar field that has a gradient in space-time with clear physical meaning: the four-current density field. The mathematical formalism of space-time Clifford algebra is introduced and then used to encode Maxwell’s equations starting only from the electromagnetic four-potential. This approach suggests a particular Zitterbewegung (ZBW) model for charged elementary particles. A straightforward application of Occam’s razor principle to Maxwell’s equation shows that only one entity, the electromagnetic four-potential, is at the origin of a plurality of concepts and entities in physics. The application of the so called "Lorenz gauge" in Maxwell's equations denies the status of real physical entity to a scalar field that has a gradient in space-time with clear physical meaning: the four-current density field. The mathematical formalism of space-time Clifford algebra is introduced and then used to encode Maxwell’s equations starting only from the electromagnetic four-potential. This approach suggests a particular Zitterbewegung (ZBW) model for charged elementary particles.
 +
 +  * [[https://hal-upec-upem.archives-ouvertes.fr/hal-01510078/document|A Geometric Algebra Implementation using Binary Tree]] (2017) - //Stéphane Breuils, Vincent Nozick, Laurent Fuchs//
 +This paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension. The method is compared with the main state of the art geometric algebra implementations, with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations.
  
   * [[https://arxiv.org/pdf/1712.05204.pdf|Inverse of multivector: Beyond p+q=5 threshold]] (2018) - //A. Acus, A. Dargys//   * [[https://arxiv.org/pdf/1712.05204.pdf|Inverse of multivector: Beyond p+q=5 threshold]] (2018) - //A. Acus, A. Dargys//
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 We consider several parameterizations of rigid transformations using motors in 3-D conformal geometric algebra. In particular, we present parameterizations based on the exponential, outer exponential, and Cayley maps of bivectors, as well as a map based on a first-order approximation of the exponential followed by orthogonal projection onto the group manifold. We relate these parameterizations to the matrix representations of rigid transformations in the 3-D special Euclidean group. Moreover, we present how these maps can be used to form retraction maps for use in manifold optimization; retractions being approximations of the exponential map that preserve the convergence properties of the optimization method while being less computationally expensive, and, for the presented maps, also easier to implement. We consider several parameterizations of rigid transformations using motors in 3-D conformal geometric algebra. In particular, we present parameterizations based on the exponential, outer exponential, and Cayley maps of bivectors, as well as a map based on a first-order approximation of the exponential followed by orthogonal projection onto the group manifold. We relate these parameterizations to the matrix representations of rigid transformations in the 3-D special Euclidean group. Moreover, we present how these maps can be used to form retraction maps for use in manifold optimization; retractions being approximations of the exponential map that preserve the convergence properties of the optimization method while being less computationally expensive, and, for the presented maps, also easier to implement.
  
-  * [[https://link.springer.com/content/pdf/10.1007%2Fs00023-018-0667-5.pdf|Generalised Spin Structures in General +  * [[https://link.springer.com/content/pdf/10.1007%2Fs00023-018-0667-5.pdf|Generalised Spin Structures in General Relativity]] (2018) - //Bas Janssens//
-Relativity]] (2018) - //Bas Janssens//+
 Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalized spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math Phys 72:103–118, 1980). Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalized spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math Phys 72:103–118, 1980).
  
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   * [[https://arxiv.org/pdf/1707.05198.pdf|On the physical interpretation of the Dirac wavefunction]] (2018) - //Anastasios Y. Papaioannou//   * [[https://arxiv.org/pdf/1707.05198.pdf|On the physical interpretation of the Dirac wavefunction]] (2018) - //Anastasios Y. Papaioannou//
 Using the language of the Geometric Algebra, we recast the massless Dirac bispinor as a set of Lorentz scalar, bivector, and pseudoscalar fields that obey a generalized form of Maxwell's equations of electromagnetism. The spinor's unusual 4-pi rotation symmetry is seen to be a mathematical artifact of the projection of these fields onto an abstract vector space, and not a physical property of the dynamical fields themselves. We also find a deeper understanding of the spin angular momentum and other Dirac field bilinears in terms of these fields and their corresponding analogues in classical electromagnetism. Using the language of the Geometric Algebra, we recast the massless Dirac bispinor as a set of Lorentz scalar, bivector, and pseudoscalar fields that obey a generalized form of Maxwell's equations of electromagnetism. The spinor's unusual 4-pi rotation symmetry is seen to be a mathematical artifact of the projection of these fields onto an abstract vector space, and not a physical property of the dynamical fields themselves. We also find a deeper understanding of the spin angular momentum and other Dirac field bilinears in terms of these fields and their corresponding analogues in classical electromagnetism.
 +
 +  * [[https://arxiv.org/pdf/1801.06054.pdf|The Four Dimensional Dirac Equation in Five Dimensions]] (2018) - //Romulus Breban//
 +The Dirac equation may be thought as originating from a theory of five-dimensional (5D) space-time. We define a special 5D Clifford algebra and introduce a spin-1/2 constraint equation to describe null propagation in a 5D space-time manifold. We explain how the 5D null formalism breaks down to four dimensions to recover two single-particle theories. Namely, we obtain Dirac's relativistic quantum mechanics and a formulation of statistical mechanics. Exploring the non-relativistic limit in five and four dimensions, we identify a new spin-electric interaction with possible applications to magnetic resonance spectroscopy (within quantum mechanics) and superconductivity (within statistical mechanics).
 +
 +  * [[https://arxiv.org/pdf/1804.04379v2.pdf|The Alternative Clifford Algebra of a Ternary Quadratic Form]] (2018) - //Adam Chapman, Uzi Vishne//
 +We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion algebra over the ring of polynomials in one variable.
 +
 +  * [[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8326692|Joint Sparse Representation Model for Multi-Channel Image Based on Reduced Geometric Algebra]] (2018) - //Miaomiao Shen, Rui Wang, Wenming Cao//
 +Sparse representations have been extended to color image processing. However, existing sparse models treat each color image pixel either as a scalar which loses color structures or as a quaternion vector matrix with high computational complexity. In this paper, we propose a novel sparse representation model for color image that bears multiple channels based on geometric algebra. First, a novel theory of reduced geometric algebra (RGA) is provided, including commutative sparse basis and the geometric operations. Second, taking advantage of the RGA theory, the model represents color image with three-channel as a multivector with the spatial and spectral information in RGA space. Third, the dictionary learning algorithm is provided using the K-RGA-based singular value decomposition (K-RGASVD) (generalized K-means clustering for RGASVD) method. The comparison results demonstrate the proposed model can remove the data redundancy and reduce the computational complexity, and can meanwhile effectively preserve the inherent color structures. The result suggests its potential as a homogeneous and efficient tool in various applications of color image analysis.
 +
 +  * [[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8322135|Multimodal Medical Image Registration Based on Feature Spheres in Geometric Algebra]] (2018) - //Wenming Cao, Fangfang Lyu et al.//
 +Multi-modal image registration in medical image analysis is very challenging as the appearance of body structures in images from different imaging devices can be very different. In this paper, we propose a new method [GA-speeded up robust features (SURF)], which incorporates the geometric algebra (GA) into SURF framework, to detect features from images. We model the volumetric data and register the multimodal medical images using feature spheres formulated in conformal geometric algebra (CGA). Specifically, we first extract features from medical images using GA-SURF. Second, we construct the feature spheres using the feature points and find the correspondence of feature spheres in the two images using CGA. With that, we can register the images based on the correspondence of the feature spheres. The experimental results evaluated by RIRE have shown that our method can register the multi-modal images with high accuracy. The maximum registration error is less than 4mm.
 +
 +  * [[https://patentimages.storage.googleapis.com/ed/27/9d/5acb66132a0753/US20180041481A1.pdf| Methods and systems for enhanced data-centric encryption systems using geometric algebra]] (2018) - //Carlos A. Paz de Araujo (X-Logos LLC)//
 +Disclosed are methods and systems to encrypt and decrypt a data message using Geometric Algebra. The encrypt operation performed on a source computing device uses the geometric product (Clifford Product) of a multivector created from plain text/data of the data message with one or more other multivectors that carry encryption keys, the identity of the source and/or other data-centric information. The source computing device sends the encrypted message to a destination computing device. The decrypt operation performed on the destination computing devices recovers the original message multivector, and ultimately the original data message by employing geometric algebra operations such as multivector inverse, Clifford conjugate and others along with the geometric product.
 +
 +  * [[https://www.researchgate.net/profile/Ying_Qiu_Gu/publication/323166788_A_Note_on_the_Representation_of_Clifford_Algebra/links/5a83ea0c45851504fb3aab00/A-Note-on-the-Representation-of-Clifford-Algebra.pdf|A Note on the Representation of Clifford Algebra]] (2018) - //Ying-Qiu Gu//
 +We construct the explicit complex and real matrix representations for the generators of real Clifford algebra Cl(p,q). The representation is based on the Pauli matrices and has an elegant structure similar to the fractal geometry, which has a period of 4. We find two class of representation, the normal representation and exceptional one. The normal representation can be expand as generators of the next 4 period, but the exceptional one can only expand at 4m+1 dimension. These results may be helpful for theoretical analysis and practical calculation.
 +
 +  * [[https://www.researchgate.net/profile/Ying_Qiu_Gu/publication/325426748_Space-Time_Geometry_and_Some_Applications_of_Clifford_Algebra_in_Physics/links/5b0d8396a6fdcc809958f409/Space-Time-Geometry-and-Some-Applications-of-Clifford-Algebra-in-Physics.pdf|Space-Time Geometry and Some Applications of Clifford Algebra in Physics]] (2018) - //Ying-Qiu Gu//
 +Directly taking the generators of Clifford algebra as tetrad of space-time, we redefine some concepts of field and then discuss the dynamical equation and symmetry by Clifford calculus. Clifford algebra exactly reflects intrinsic symmetry of fields with no more or less contents, and automatically classifies the parameters in field equation by grade, which is a definite guidance to set up dynamical equation and compatible constraints of fields. By insights of Clifford algebra, we discuss the connection of fields and torsion in details. The dynamical equation of torsion, the quadratic form of Lagrangian of gravity and first order dynamical equation of tensors in curved space-time are derived. By virtue of this excellent language, physical theories can be well understood by common readers, and some long standing puzzling problems may be easily solved.
 +
 +  * [[https://arxiv.org/pdf/1803.06194.pdf|Factorization of Left Polynomials in Clifford Algebras: State of the Art, Applications, and Open Questions]] (2018) - //Zijia Li, Hans-Peter Schröcker//
 +We discuss existence of factorizations with linear factors for (left) polynomials over rings with real center. Important examples come from Clifford algebra. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of factorizations for generic polynomials over certain division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples.
 +
 +  * [[https://arxiv.org/pdf/1805.03126.pdf|On complex representations of Clifford algebra]] (2018) - //Marco Budinich//
 +We show that complex representations of Clifford algebra can always be reduced either to a real or to a quaternionic algebra depending on signature of complex space thus showing that complex spinors are unavoidably either real Majorana spinors or quaternionic spinors. We use this result to support (1,3) signature for Minkowski space.
 +
 +  * [[http://www.mdpi.com/2220-9964/7/5/172|Optimal Route Searching with Multiple Dynamical Constraints - A Geometric Algebra Approach]] (2018) - //Dongshuang Li et al.//
 +The process of searching for a dynamic constrained optimal path has received increasing attention in traffic planning, evacuation, and personalized or collaborative traffic service. As most existing multiple constrained optimal path (MCOP) methods cannot search for a path given various types of constraints that dynamically change during the search, few approaches for dynamic multiple constrained optimal path (DMCOP) with type II dynamics are available for practical use. In this study, we develop a method to solve the DMCOP problem with type II dynamics based on the unification of various types of constraints under a geometric algebra (GA) framework. In our method, the network topology and three different types of constraints are represented by using algebraic base coding. With a parameterized optimization of the MCOP algorithm based on a greedy search strategy under the generation-refinement paradigm, this algorithm is found to accurately support the discovery of optimal paths as the constraints of numerical values, nodes, and route structure types are dynamically added to the network.
 +
 +  * [[https://www.researchgate.net/profile/Bernd_Schmeikal/publication/324685162_Directional_Domains_of_Physical_Motion-Einstein%27s_Blunder/links/5adc7744458515c60f5e1fb0/Directional-Domains-of-Physical-Motion-Einsteins-Blunder.pdf|Directional Domains of Physical Motion - Einstein's Blunder]] (2018) - //Bernd Schmeikal//
 +Einstein's blunder was to take over Minkowski's World Point and World Line concepts. The early treatises on Relativity and Relativity Principle contained an error concerning rotation of so called rigid bodies and some invalid concept of world and worldline. The theory showed a clearly recognizable paradox, which had enormous consequences for the mathematical execution of its layout. If we remove those errors, Relativity and Quantum Theory come closer. This required at least to provide Relativity Theory with extended algebraic means, and to define a better concept of the directional domain of quantum motion in the algebraic basis. Once this is carried out, we come up with a better understanding of matter and space-time that involves causality amending, the interaction of fermion boosts, warps, enhancement and shrinking of space-time-volumes. Local Four-volumes, -as directors of the Clifford algebra of the Minkowski space-time,- are no longer invariant pseudoscalars like in the old theory, but are converted into time.
 +
 +  * [[http://tuprints.ulb.tu-darmstadt.de/7232/13/TUDthesisFromMath.pdf|Space-Time Discretization of Maxwell's Equations in the Setting of Geometric Algebra]] (2018) - //Mariusz Klimek//
 +In this dissertation a FIT-like discretization of Maxwell's equations is performed directly in four-dimensional space-time using the mathematical formalism of Clifford's Geometric Algebra
 +
 +  * [[https://web.archive.org/web/20180821225013/http://www.kurtnalty.com/RegressiveVsAntiWedge.pdf|Regressive Versus Antiwedge Products]] (2018) - //Kurt Nalty//
 +The Hestenes (1986) regressive product differs from the Lengyel antiwedge product.
 +
 +  * [[http://www.kurtnalty.com/Dets_Idem_and_Nil_Potents.pdf|Determinants, Idempotents and Nilpotents in Geometric Algebra]] (2018) - //Kurt Nalty//
 +I present formulas for determinants, nilpotents (non-zero expressions which square to zero) and idempotents (expressions which square to themselves) in two through five dimensional Geometric Algebras. Formulas implemented using nilpotents and idempotents are simple, and aggressively linear.
 +
 +  * [[https://www.math.ubc.ca/~cass/research/pdf/Clifford.pdf|Clifford algebras and spinors]] (2018) - //Bill Casselman//
 +This essay will present a brief outline of the theory of Clifford algebras, together with a small amount of material about quadratic forms. I follow loosely the well known book Geometric Algebra by Emil Artin, but with elegant modifications that I saw originally in some lecture notes by Raoul Bott (dating from 1962), subsequently included in Atiyah-Bott-Shapiro (1964).
 +
 +  * [[https://jivp-eurasipjournals.springeropen.com/track/pdf/10.1186/s13640-018-0318-2|A conformal geometric algebra method for virtual hand modeling and interaction]] (2018) - //Peng Chen, Yong Hu, Fengwei Yang//
 +Virtual hand usually simulates human hands by mapping the actual shapes of hands to virtual environment and modeling the hand-object manipulations in human machine interactions. For describing virtual hand and its manipulation in a unified framework, a method based on conformal geometric algebra (CGA) is proposed to solve the problems of virtual hand modeling and interaction in this paper. With the vertex blending based on CGA, the artifacts on the finger joints are improved, which enhances the realism of the virtual hand model. With the same tool, the collision detection between the virtual hand and the manipulated objects is implemented. The gesture of grasp, pinch, and hold are recognized, and the corresponding manipulation rules are established by CGA calculation.
 +
 +  * [[https://arxiv.org/pdf/1808.02363|Geometric Matrices and the Symmetric Group]] (2018) - //Garret Sobczyk//
 +We construct real and complex matrices in terms of Kronecker products of a Witt basis of 2n null vectors in the geometric algebra over the real and complex numbers. In this basis, every matrix is represented by a unique sum of products of null vectors. The complex matrices provide a direct matrix representation for geometric algebras with signatures p+q ≤ 2n+1. Properties of irreducible representations of the symmetric group are presented in this geometric setting.
 +
 +  * [[https://dspace.library.uu.nl/bitstream/handle/1874/366510/thesis.pdf?sequence=2&isAllowed=y|The Clifford algebra and its Spin subgroup]] (2018) - //Pepijn de Maat//
 +This thesis gives an introduction to the Clifford algebra and its Spin subgroup. We first construct the Clifford algebra before using the spinorial representation to proof its (semi)simplicity. We then construct the Spin group and determine its irreducible representations using weight theory.
 +
 +  * [[https://arxiv.org/pdf/1808.06240|Multisymplectic structures and invariant tensors for Lie systems]] (2018) - //Xavier Gràcia et al.//
 +A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory.
 +
 +  * [[http://afrodita.zcu.cz/~skala/PUBL/Publ_2018/2018-Plucker-MCSI-Skala-Smolik.pdf|A New Formulation of 
 +Plücker Coordinates Using Projective Representation]] (2018) - //Vaclav Skala, Michal Smolik//
 +This contribution presents a new formulation of Plücker coordinates using geometric algebra and standard linear algebra with projective representation. The Plücker coordinates are usually used for a line representation in space, which is given by two points. However, the line can be also given as an intersection of two planes in space. The principle of duality leads to a simple formulation for both cases.The presented approach uses homogeneous coordinates with the duality principle application. It is convenient for application on GPU as well.
 +
 +  * [[https://www.researchgate.net/profile/Debashis_Sen/publication/327262811_Geometric_Algebra_as_the_unified_mathematical_language_of_Physics_An_introduction_for_advanced_undergraduate_students/links/5b8f6e8fa6fdcc1ddd0fea28/Geometric-Algebra-as-the-unified-mathematical-language-of-Physics-An-introduction-for-advanced-undergraduate-students.pdf|Geometric Algebra as the unified mathematical language of Physics: An introduction for advanced undergraduate students]] (2018) - //Debashis Sen, Deeprodyuti Sen//
 +In recent years, geometric algebra has emerged as the preferred mathematical framework for physics. It provides both compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. Geometric algebra has also found prolific applications as a computational tool in computer graphics and robotics. Leading exponents of this extensive mathematical apparatus are fervently insisting its inclusion in the undergraduate physics curriculum and in this paper an introductory exposure, in familiar terms for the advanced undergraduate students, is intended.
 +
 +  * [[https://pastel.archives-ouvertes.fr/tel-02085820/document|Algorithmic structure for geometric algebra operators and application to quadric surfaces]] (2018) - //Stephane Breuils//
 +Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics problems. The Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to the high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. In this thesis, we first present a Geometric Algebra implementation dedicated for both low and high dimensions.
 +
 +  * [[https://arxiv.org/pdf/1809.09706|Notes on Plucker's relations in Geometric Algebra]] (2018) - //Garret Sobczyk//
 +Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the basic idea of the Plucker relations in Clifford's geometric algebra. We discover that the Plucker Relations can be fully characterized in terms of the geometric product. 
 +
 +  * [[https://arxiv.org/pdf/1908.08110.pdf|On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space]] (2019) - //Jaime Vaz, Stephen Mann//
 +We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra Cl(3,3) of the quadratic space R(3,3). We show that this algebra describes in a unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using the concept of Hodge duality, we define an operation called cotranslation, and show that the operation of perspective projection can be written in this Clifford algebra as a composition of the translation and cotranslation operations. We also show that the operation of pseudo-perspective can be implemented using the cotranslation operation. 
 +
 +  * [[https://arxiv.org/pdf/1908.04590|Real spinors and real Dirac equation]] (2019) - //Vaclav Zatloukal//
 +We reexamine the minimal coupling procedure in the Hestenes' geometric algebra formulation of the Dirac equation, where spinors are identified with the even elements of the real Clifford algebra of spacetime. This point of view, as we argue, leads naturally to a non-Abelian generalisation of the electromagnetic gauge potential.
 +
 +  * [[https://arxiv.org/pdf/1908.02235.pdf|Real Clifford algebras and their spinors for relativistic fermions]] (2019) - //Stefan Floerchinger//
 +Real Clifford algebras for arbitrary number of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relation between mathematical structures and applications to describe relativistic fermions is emphasized throughout. 
 +
 +  * [[https://link.springer.com/content/pdf/10.1007%2Fs00006-019-0991-y.pdf|Geometric Algebra, Gravity and Gravitational Waves]] (2019) - //Anthony Lasenby//
 +We discuss an approach to gravitational waves based on Geometric Algebra and Gauge Theory Gravity. After a brief introduction to Geometric Algebra (GA), we consider Gauge Theory Gravity, which uses symmetries expressed within the GA of flat spacetime to derive gravitational forces as the gauge forces corresponding to making these symmetries local. We then consider solutions for black holes and plane gravitational waves in this approach, noting the simplicity that GA affords in both writing the solutions, and checking some of their properties. We then go on to show that a preferred gauge emerges for gravitational plane waves, in which a ‘memory effect’ corresponding to non-zero velocities left after the passage of the waves becomes clear, and the physical nature of this effect is demonstrated. In a final section we present the mathematical details of the gravitational wave treatment in GA, and link it with other approaches to exact waves in the literature.
 +
 +  * [[https://arxiv.org/pdf/1906.11622.pdf|On the generalized spinor field classification: Beyond the Lounesto Classification]] (2019) - //C. H. Coronado Villalobos et al.//
 +In this paper we advance into a generalized spinor field classification, based on the so-called Lounesto classification. The program developed here is based on an existing freedom on the spinorial dual structures definition, which, in certain simple physical and mathematical limit, allows us to recover the usual Lounesto classification. The protocol to be accomplished give full consideration in the understanding of the underlying mathematical structure, in order to satisfy the quadratic algebraic relations known as Fierz-Pauli-Kofink identities, and also to provide physical observables. As we will see, such identities impose a given restriction on the number of possible spinor field classes in the classification. We also expose a mathematical device known as Clifford algebra deformation, which ensures real spinorial densities and holds the Fierz-Pauli-Kofink quadratic relations. 
 +
 +  * [[https://www.tandfonline.com/doi/pdf/10.1080/19475683.2019.1612945?needAccess=true|Towards the next-generation GIS: a geometric algebra approach]] (2019) - //Linwang Yuan, Zhaoyuan Yu, Wen Luo//
 +(...) Geometric algebra (GA) provides an ideal tool for the expression and calculation of multidimensional geometric objects, and has proved to be effective for GIS representation and computation applications in our previous studies. We propose to use GA as the basic mathematical language for the establishment of the next-generation GIS. We present the framework of a GA-based next-generation GIS and describe the representation space, data structure, and computational models in this paper. A few issues that have not been sufficiently addressed by previous studies are discussed in detail with potential solutions proposed. 
 +
 +  * [[https://odr.chalmers.se/bitstream/20.500.12380/257390/1/257390.pdf|Rotationer, spinorer och spinn]] (2019) - //Patrik Agné, Simon Jonsson, Xu Liqin, Lars Wickström//
 +Ett viktigt begrepp inom kvantmekaniken är spinn. Vissa kvantmekaniska system har egenskapen att vid en full rotation har systemet inte återställts utan befinner sig istället i motsattkonfiguration relativt startläget. Detta är vad man menar med spinn. Spinn är dock känt föratt vara svårt att visualisera. I detta arbete har vi skapat en datoranimation för att visa hurspinn uppkommer och beter sig. Vi har använt programspråket MATLAB för att göra detta.För att kunna förstå denna datoranimation måste man dock först ha grundläggande förståelseför spinn. I detta arbete har vi därför gjort en genomgång av den matematiska teorin bakomspinn. Vi börjar med att förklara begreppen yttre algebra och Cliffordalgebra. Sedan introducerar vi kvaternioner och förklarar deras koppling till spinn. Vi går därefter igenom begreppenspinorer och spinorrum som är nödvändiga för att beskriva spinn i fysiken. Vi avslutar arbetetmed att förklara hur koden är uppbyggd och hur den är kopplad till spin.
 +
 +  * [[https://link.springer.com/content/pdf/10.1007%2Fs00006-019-0960-5.pdf|Conformal Villarceau Rotors]] (2019) - //Leo Dorst//
 +We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori. We compute the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point. This allows compact derivation of the quantitative symmetry properties of the Villarceau circles. We briefly derive their role in the Hopf fibration and as stereographic images of isoclinic rotations on a 3-sphere of the 4D Clifford torus. We use the CGA description to generate 3D images of our results, by means of GAviewer. This paper was motivated by the hope that the compact coordinate-free CGA representations can aid in the analysis of Villarceau circles (and torus knots) as occurring in the Maxwell and Dirac equations.
 +
 +  * [[http://www.dpi-proceedings.com/index.php/dtcse/article/download/30081/28960|Human Joint Orientation Descriptor Based on Geometric Algebra and Its Application]] (2019) - //Wen-ming Cao, Yi-tao Lu//
 +Motion recognition is becoming more and more widely used in various applications. In this paper we propose a novel descriptor to describe human skeleton based on geometric algebra (GA) that decomposes the skeleton posture into the rotations of skeleton parts. In this model, all body bones are rotated from the same original states. We formulate the rotation operator in 3D GA space, which can be used to describe the rotations of human body bones. Then we select the most informative rotations of body bones and joint angles to represent the skeleton. We train a Gaussian Naïve Bayes classifier which can recognize the motion type of a single input frame captured from video sensors. After the motion type is determined, we find the most similar posture in the motion sequence database using the distance based on posture orientations and joint angles. And finally, we calculate the posture difference to give users the calibration advice. Our experimental results have shown the high accuracy and effectiveness of our method.
 +
 +  * [[https://arxiv.org/pdf/1904.00084.pdf|A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations]] (2019) - //Dimiter Prodanov//
 +Clifford algebras are an active area of mathematical research. The main objective of the paper is to exhibit a construction of a matrix algebra isomorphic to a Clifford algebra of signature (p,q), which can be automatically implemented using general purpose linear algebra software. While this is not the most economical way of implementation for lower-dimensional algebras it offers a transparent mechanism of translation between a Clifford algebra and its isomorphic faithful real matrix representation. Examples of lower dimensional Clifford algebras are presented.
 +
 +  * [[https://www.researchgate.net/publication/331531539_A_Spinor_Model_for_Cascading_Two_Port_Networks_In_Conformal_Geometric_Algebra|A Spinor Model for Cascading Two Port Networks In Conformal Geometric Algebra]] (2019) - //Alex Arsenovic//
 +This paper shows how Conformal Geometric Algebra (CGA) can be used to model an arbitrary two-port network as a rotation in four dimensional space, known as a spinor. This spinor model plays the role of the wave-cascading matrix in conventional network theory. Techniques to translate two-port scattering matrix in and out of spinor form are given. Once the translation is laid out, geometric interpretations are given to the physical properties of reciprocity, loss, and symmetry and some mathematical groups are identified. Methods to decompose a network into various sub-networks, are given. Since rotations in four dimensional Minkowski space are Lorentz transformations, our model opens up up the field of network theory to physicists familiar with relativity, and vice versa. The major drawback to the approach is that Geometric Algebra is relatively unknown. However, it is precisely the Geometric Algebra which provides the insight and universality of the model.
 +
 +  * [[http://ijeee.iust.ac.ir/article-1-1312-en.pdf|Clifford Algebra's Geometric Product Properties in Image-Processing and its Efficient Implementation]] (2019) - //Ali Sadr, Niloofar Orouji//
 +Clifford Algebra (CA) is an effective substitute for classic algebra as the modern generation of mathematics. However, massive computational loads of CA-based algorithms have hindered its practical usage in the past decades. Nowadays, due to magnificent developments in computational architectures and systems, CA framework plays a vital role in the intuitive description of many scientific issues. Geometric Product is the most important CA operator, which created a novel perspective on image processing problems. In this work, Geometric Product and its properties are discussed precisely, and it is used for image partitioning as a straightforward instance. Efficient implementation of CA operators needs a specialized structure, therefore a hardware architecture is proposed that achieves 25x speed-up in comparison to the software approach.
 +
 +  * [[https://digital-library.theiet.org/content/journals/10.1049/joe.2019.0048|Theorems of compensation and Tellegen in non-sinusoidal circuits via geometric algebra]] (2019) - // Milton Castro-Núñez, Deysy Londoño-Monsalve, Róbinson Castro-Puche//
 +Presently, it is not possible to corroborate Tellegen's theorem or to articulate the compensation theorem in the frequency domain when considering all the harmonics simultaneously. The circuit analysis approach based on geometric algebra is used here to solve these two challenges. We show here the significance of representing harmonics by k-vectors and how k-vectors process the magnitude, the phase and the frequency of a sine wave. We take a tutorial approach and provide examples to demonstrate both, the simplicity of this approach and how a distinct representation of time-domain signals of different frequencies facilitates both, energy analysis and confirming the principle of superposition and Kirchhoff's circuits' laws in non-sinusoidal conditions when considering all the harmonics simultaneously.
 +
 +  * [[https://arxiv.org/pdf/1902.05478.pdf|A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks]] (2019) - //Fidelis Zanetti de Castro, Marcos Eduardo Valle//
 +In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras.
 +
 +  * [[https://arxiv.org/pdf/1903.02444|Efficient representation and manipulation of quadratic surfaces using Geometric Algebras]] (2019) - //Stéphane Breuils, Vincent Nozick, Laurent Fuchs, Akihiro Sugimoto//
 +Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these frameworks support all the operations required to completely handle these surfaces. Some frameworks do not allow the construction of quadratic surfaces from control points when others do not allow to transform these quadratic surfaces. However , if we consider all the frameworks together, then all the required operations over quadratic are covered. This paper presents a unification of these frameworks that enables to represent any quadratic surfaces either using control points or from the coefficients of its implicit form. The proposed approach also allows to transform any quadratic surfaces and to compute their intersection and to easily extract some geometric properties. 
 +
 +  * [[https://arxiv.org/pdf/1901.05873.pdf|Projective geometric algebra: A new framework for doing euclidean geometry]] (2019) - //Charles G. Gunn//
 +A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior algebras as sub-algebras simplifies the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity.
 +
 +  * [[https://arxiv.org/pdf/1909.02408|A Low-Memory Time-Efficient Implementation of Outermorphisms for Higher-Dimensional Geometric Algebras]] (2019) - //Ahmad Hosny Eid//
 +Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. (...) This work attempts to shed some light on the problem of optimizing software implementations of outermorphisms for practical prototyping applications using geometric algebra. The approach we propose here for implementing outermorphisms requires orders of magnitude less memory compared to other common approaches, while being comparable in time performance, especially for high-dimensional geometric algebras.
 +
 +  * [[http://downloads.hindawi.com/journals/cin/2019/9374802.pdf|Evaluating a Semi-Autonomous Brain-Computer Interface Based on Conformal Geometric Algebra and Artificial Vision]] (2019) - //Mauricio Ramirez-Moreno, David Gutiérrez//
 +We evaluate a semi-autonomous brain-computer interface (BCI) for manipulation tasks. In such system, the user controls a robotic arm through motor imagery commands. (...) We take a semi-autonomous approach based on a conformal geometric algebra model that solves the inverse kinematics of the robot on the fly, then the user only has to decide on the start of the movement and the final position of the effector (goal-selection approach). Under these conditions, we implemented pick-and-place tasks with a disk as an item and two target areas placed on the table at arbitrary positions.
 +
 +  * [[https://arxiv.org/pdf/1912.11198|Geometric Obstructions on Gravity]] (2019) - //Yuri Martins, Rodney Biezuner//
 +These are notes for a short course and some talks gave at Departament of Mathematics and at Departament of Physics of Federal University of Minas Gerais, based on the author's paper. (...) We present obstructions to realize gravity, modeled by the tetradic Einstein-Hilbert-Palatini (EHP) action functional, in a general geometric setting.
 +
 +  * [[https://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez//
 +Robust methods for finding the best rotation aligning two sets of corresponding vectors are formulated in the linear algebra framework, using tools like the SVD for polar decomposition or QR for finding eigenvectors. Those are well established numerical algorithms which on the other hand are iterative and computationally expensive. Recently, closed form solutions has been proposed in the quaternion’s framework, those methods are fast but they have singularities i.e., they completely fail on certain input data. In this paper we propose a robust attitude estimator based on a formulation of the problem in Geometric Algebra. We find the optimal eigen-quaternion in closed form with high accuracy and with competitive performance respect to the fastest methods reported in literature.
 +
 +  * [[https://arxiv.org/pdf/1911.08658|Spinors of real type as polyforms and the generalized Killing equation]] (2019) - //Vicente Cortes, Calin Lazaroiu, C. S. Shahbazi//
 +We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS_4 space-time.
 +
 +  * [[https://link.springer.com/content/pdf/10.1007/s00006-019-0987-7.pdf|Garamon: A Geometric Algebra Library Generator]] (2019) - //Stephane Breuils, Vincent Nozick, Laurent Fuchs//
 +This paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++/Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. 
 +
 +  * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler//
 +This is the first entry in a planned series aiming to establish a modified, and simpler, formalism for studying the geometry of smooth manifolds with a metric, while remaining close to standard textbook treatments in terms of notation and concepts. The key step is extending the tangent space at each point from a vector space to a geometric algebra, which is a linear space incorporating vectors with dot and wedge multiplication, and extending the affine connection to a directional derivative acting naturally on fields of multivectors (elements of the geometric algebra). (...) The theory that results from this extension is simpler and more powerful than either differential forms or tensor methods, in a number of ways.
 +
 +  * [[https://arxiv.org/pdf/2002.11313|Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors]] (2020) - //Stephane Breuils, Vincent Nozick, Akihiro Sugimoto//
 +Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it does not reflect the practical usage of geometric algebra. Indeed, in applications related to geometry, multivectors are likely to be full homogeneous, having their non-zero elements over a single grade. In this paper, we provide a complete computational study on geometric algebra products of two full homogeneous multivectors, that is, the outer, inner, and geometric products of two full homogeneous multivectors. We show tight bounds on the number of the arithmetic operations required for these products.
 +
 +  * [[https://arxiv.org/pdf/2001.00656|Two-State Quantum Systems Revisited: A Geometric Algebra Approach]] (2020) - //Pedro Amao, Hernán Castillo//
 +We revisit the topic of two-state quantum systems using Geometric Algebra (GA) in three dimensions G3. In this description, both the quantum states and Hermitian operators are written as elements of G3. By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in GA. We then use this approach to revisit the problem of a spin-1/2 particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, GA reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of G3.
 +
 +  * [[https://arxiv.org/abs/2003.07159|Periodic Table of Geometric Numbers]] (2020) - //Garret Sobczyk//
 +Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in mathematics is the Periodic Table of Geometric Numbers discussed here. In 1878 William Kingdon Clifford discovered the defining rules for what he called geometric algebras. We show how these algebras, and their coordinate isomorphic geometric matrix algebras, fall into a natural periodic table, sidelining the superfluous definitions based upon tensor algebras and quadratic forms.
 +
 +  * [[https://arxiv.org/pdf/2002.05993|Projective Geometric Algebra as a Subalgebra of Conformal Geometric Algebra]] (2020) - //Jaroslav Hrdina, Ales Navrat, Petr Vasik, Dietmar Hildenbrand//
 +First we introduce briefly the frameworks of CGA and PGA for doing Euclidean geometry and we summarise basic formulas. In the next section, we show that there are actually two naturally related copies of PGA in CGA. After an identification of the two copies, the duality in PGA is obtained in terms of CGA operations. This implies directly the correspondence between flat objects and versors for Euclidean transformations in CGA and the objects and versors in PGA.
 +
 +  * [[https://link.springer.com/content/pdf/10.1007/s00006-020-1046-0.pdf|A 1d Up Approach to Conformal Geometric Algebra]] (2020) - //Anthony N. Lasenby//
 +We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell's Theorem in quantum mechanics, and its relation to the sphere S^7 and the exceptional group E_8, and proposed a new associative version of the division algebra normally thought to require the octonians.
 +
 +  * [[https://arxiv.org/pdf/2004.06655|Dimensional scaffolding of electromagnetism using geometric algebra]] (2020) - //Xabier Prado Orbán, Jorge Mira//
 +Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. (...) Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism.
 +
 +  * [[https://www.ram-lab.com/papers/2020/wu2020iet.pdf|A Linear Geometric Algebra Rotor Estimator for Efficient Mesh Deformation]] (2020) - //Jin Wu, Mauricio Lopez, Ming Liu, Yilong Zhu//
 +We solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. We show its efficiency in two applications: the As-Rigid-As-Possible (ARAP) Surface Modeling and the more Smooth Rotation enhanced As-Rigid-As-Possible (SR-ARAP) mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics.
 +
 +  * [[http://publikacio.uni-eszterhazy.hu/5004/1/AMI_online_1059.pdf|Optimized line and line segment clipping in E2 and Geometric Algebra]] (2020) - //Vaclav Skala//
 +Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the E^2 or E^3 space. The projection operation from the E^3 to the E^2 space leads to the necessity to convert coordinates to the Euclidean space if the clipping operation is to be used. In this contribution, an optimized simple algorithm for line and line segment clipping in the E^2 space, which works directly with homogeneous representation and not requiring the conversion to the Euclidean space, is described. It is based on Geometric Algebra (GA) formulation for projective representation. The proposed algorithm is simple, efficient and easy to implement.
 +
 +  * [[https://arxiv.org/pdf/2007.04464.pdf|Deform, Cut and Tear a skinned model using Conformal Geometric Algebra]] (2020) - //Manos Kamarianakis, George Papagiannakis//
 +We present a novel, integrated rigged character simulation framework in Conformal Geometric Algebra (CGA) that supports, for the first time, real-time cuts and tears, before and/or after the animation, while maintaining deformation topology. The purpose of using CGA is to lift several restrictions posed by current state-of-the-art character animation & deformation methods. Previous implementations originally required weighted matrices to perform deformations, whereas, in the current state-of-the-art, dual-quaternions handle both rotations and translations, but cannot handle dilations. CGA is a suitable extension of dual-quaternion algebra that amends these two major previous shortcomings: the need to constantly transmute between matrices and dual-quaternions as well as the inability to properly dilate a model during animation. Our CGA algorithm also provides easy interpolation and application of all deformations in each intermediate steps, all within the same geometric framework. Furthermore we also present two novel algorithms that enable cutting and tearing of the input rigged, animated model, while the output model can be further re-deformed.
 +
 +  * [[https://www.researchgate.net/profile/Carlos-Muro/publication/339249543_Newton-Euler_Modeling_and_Control_of_a_Multi-copter_Using_Motor_Algebra_mathbfG_301G301/links/603d925f299bf1e0784d02bd/Newton-Euler-Modeling-and-Control-of-a-Multi-copter-Using-Motor-Algebra-mathbfG-3-0-1G3-0-1.pdf|Newton-Euler Modelling and Control of a Multicopter using Motor Algebra G^+_3,0,1]] (2020) - //Carlos A. Arellano-Muro, Guillermo Osuna-Gonzalez, et al//
 +In this work the dynamic model and the nonlinear control for a multi-copter have been developed using the geometric algebra framework specifically using the motor algebra G^+_3,0,1. The kinematics for the aircraft model and the dynamics based on Newton-Euler formalism are presented. Block-control technique is applied to the multi-copter model which involves super twisting control and an estimator of the internal dynamics for maneuvers away from the origin. The stability of the presented control scheme is proved. The experimental analysis shows that our non-linear controller law is able to reject external disturbances and to deal with parametric variations.
 +
 +  * [[http://i-us.ru/index.php/ius/article/download/13579/14098|Human action recognition method based on conformal geometric algebra and recurrent neural network]] (2020) - //Nguyen Nang Hung Van, Pham Minh Tuan et al//
 +The use of Conformal Geometric Algebra in order to extract features and simultaneously reduce the dimensionality of a dataset for human activity recognition using Recurrent Neural Network.
 +
 +  * [[https://arxiv.org/pdf/2107.00343.pdf|Explicit Baker-Campbell-Hausdorff-Dynkin formula for Spacetime via Geometric Algebra]] (2021) - //Joseph Wilson, Matt Visser//
 +We present a compact Baker-Campbell-Hausdorff-Dynkin formula for the composition of Lorentz transformations e^σi in the spin representation (a.k.a. Lorentz rotors) in terms of their generators σi. This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension ≤4, naturally generalising Rodrigues' formula for rotations in R3. In particular, it applies to Lorentz rotors within the framework of Hestenes' spacetime algebra, and provides an efficient method for composing Lorentz generators.
 +
 +  * [[https://arxiv.org/pdf/2107.03771.pdf|Graded Symmetry Groups: Plane and Simple]] (2021) - //Martin Roelfs, Steven De Keninck//
 +The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a graded structure on Lie groups, which is not evident in their matrix representation. By embracing this graded structure, the invariant decomposition theorem was proven: any composition of k linearly independent reflections can be decomposed into ⌈k/2⌉ commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi-Chasles' theorem as its 3D Euclidean special case.
 +
 +  * [[https://www.pacm.princeton.edu/sites/default/files/pacm_arjunmani_0.pdf|Representing Words in a Geometric Algebra]] (2021) - //Arjun Mani//
 +In this paper we introduce and motivate geometric algebra as a better representation for word embeddings. Next we describe how to implement the geometric product and interestingly show that neural networks can learn this product. We then introduce a model that represents words as objects in this algebra and benchmark it on large corpuses; our results show some promise on traditional word embedding tasks. Thus, we lay the groundwork for further investigation of geometric algebra in word embeddings.
 +
 +  * [[https://ietresearch.onlinelibrary.wiley.com/doi/pdfdirect/10.1049/cmu2.12188|An approach to adaptive filtering with variable step size based on geometric algebra]] (2021) - //Haiquan Wang, Yinmei He et al//
 +Recently, adaptive filtering algorithms have attracted much more attention in the field of signal processing. By studying the shortcoming of the traditional real-valued fixed step size adaptive filtering algorithm, this paper proposed the novel approach to adaptive filtering with variable step size based on Sigmoid function and geometric algebra (GA).
 +
 +  * [[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9488174|A Survey on Quaternion Algebra and Geometric Algebra Applications in Engineering and Computer Science 1995–2020]] (2021) - //Eduardo Bayro-Corrochano//
 +Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020.
 +
 +  * [[https://dspace.library.uu.nl/bitstream/handle/1874/403340/thesis.pdf|Clifford algebras and their application in the Dirac equation]] (2021) - //Paul van Hoegaerden//
 +The aim of this thesis will be to study the Clifford algebras that appear in the derivation of the Dirac equation and investigate alternative formulations of the Dirac equation using (complex) quaternions. To this end, we will first look at the symmetries of the Dirac equation and some of the additional insights that follow from the Dirac equation. We will also give a derivation of the Dirac equation starting from the Schrödinger equation, in which we will come across the gamma matrices.
 +
 +  * [[https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/cta.3132|Geometric Algebra for teaching AC Circuit Theory]] (2021) - //Francisco G. Montoya, Raúl Baños et al//
 +This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non-sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses.
 ===== Books ===== ===== Books =====
  
 ==== Historical ==== ==== Historical ====
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 ^                                                                                                                                           ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ^ ^                                                                                                                                           ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ^
-| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?400}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |+| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?100}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |
 | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]]                                  | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.                                                                                                                                        | | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]]                                  | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.                                                                                                                                        |
 | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]]                            | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]]                            | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
Line 1115: Line 1413:
 | [[https://archive.org/details/anelementaryexp00collgoog|{{:ga:an_elementary_exposition_of_grassmann_ausdehnunglsehre-collins.jpg?100}}]]  | **An Elementary Exposition of Grassmann's Ausdehnungslehre or Theory of Extensions (1901)**\\ //Joseph V. Collins//\\ (...) Hyde, in his Directional Calculus, purposing to present the Ausdehnungslehre to American readers, cut  the knot of the difficulty by taking the results of the theoretical part for granted and giving only the application to geometry, and by limiting his treatment to two and three dimensional space.  An elementary exposition which will give the simpler portions of the theoretical part as well as the applications of the theory seems to be needed. Such an exposition should serve the needs of two classes of readers: First, of  those who would like to have a good general idea of the subject without going very deeply into its particulars; and secondly, of those who, expecting to make a thorough study of the subject, wish first to read an introduction to it.                                                                                          | | [[https://archive.org/details/anelementaryexp00collgoog|{{:ga:an_elementary_exposition_of_grassmann_ausdehnunglsehre-collins.jpg?100}}]]  | **An Elementary Exposition of Grassmann's Ausdehnungslehre or Theory of Extensions (1901)**\\ //Joseph V. Collins//\\ (...) Hyde, in his Directional Calculus, purposing to present the Ausdehnungslehre to American readers, cut  the knot of the difficulty by taking the results of the theoretical part for granted and giving only the application to geometry, and by limiting his treatment to two and three dimensional space.  An elementary exposition which will give the simpler portions of the theoretical part as well as the applications of the theory seems to be needed. Such an exposition should serve the needs of two classes of readers: First, of  those who would like to have a good general idea of the subject without going very deeply into its particulars; and secondly, of those who, expecting to make a thorough study of the subject, wish first to read an introduction to it.                                                                                          |
 | [[https://archive.org/details/117714283|{{:ga:vector_analysis-gibbs.jpg?100}}]]                                                           | **Vector Analysis (1901)**\\ //J. Willard Gibbs, E. B. Wilson//\\ Vector Analysis is a textbook by Edwin Bidwell Wilson, first published in 1901 and based on the lectures that Josiah Willard Gibbs had delivered on the subject at Yale University. The book did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus, as used by physicists and mathematicians.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | | [[https://archive.org/details/117714283|{{:ga:vector_analysis-gibbs.jpg?100}}]]                                                           | **Vector Analysis (1901)**\\ //J. Willard Gibbs, E. B. Wilson//\\ Vector Analysis is a textbook by Edwin Bidwell Wilson, first published in 1901 and based on the lectures that Josiah Willard Gibbs had delivered on the subject at Yale University. The book did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus, as used by physicists and mathematicians.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
 +| [[https://archive.org/details/constructionands033261mbp|{{:ga:the_construction_and_study_of_certain_important_algebras-chevalley.jpg?100}}]]                                                           | **The Construction and Study of Certain Important Algebras (1955)**\\ //Claude Chevalley//\\ This booklet reproduces lectures given by the author in Japan in 1954 on the main topics in multilinear algebra. But the subject is approached from an unusual point of view, the importance of which has only been realized during the last few years, mainly in the recent work on homology and Lie theory; and it is chiefly for the introduction it presents to these new methods that the book is especially valuable, being probably the first of its kind to appear in print. The main themes of the book are the notion of "universal" algebra with respect to a given property, and the notion of graded algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
 | [[https://archive.org/details/geometricalgebra033556mbp|{{:ga:geometric_algebra-artin_ip.jpg?100}}]]                                      | **Geometric Algebra (1957)**\\ //Emil Artin//\\ Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | | [[https://archive.org/details/geometricalgebra033556mbp|{{:ga:geometric_algebra-artin_ip.jpg?100}}]]                                      | **Geometric Algebra (1957)**\\ //Emil Artin//\\ Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
 | [[https://archive.org/details/cliffordnumberss00ries|{{:ga:clifford_numbers_and_spinors-riesz.jpg?100}}]]                                 | **Clifford Numbers and Spinors (1958)**\\ //Marcel Riesz//\\ The presentation of the theory of Clifford numbers and spinors given here has grown out of thirteen lectures delivered at the Institute for Fluid Dynamics and Applied Mathematics of the University of Maryland. The work is divided into six chapters which, for the convenience of those readers who are only interested in certain parts of the material treated, are largely independent of each other. This arrangement has, of course, certain disadvantages such as repetitions and over-lappings.                                                                                                                                                                                                                                                                                                                                                                                                                                       | | [[https://archive.org/details/cliffordnumberss00ries|{{:ga:clifford_numbers_and_spinors-riesz.jpg?100}}]]                                 | **Clifford Numbers and Spinors (1958)**\\ //Marcel Riesz//\\ The presentation of the theory of Clifford numbers and spinors given here has grown out of thirteen lectures delivered at the Institute for Fluid Dynamics and Applied Mathematics of the University of Maryland. The work is divided into six chapters which, for the convenience of those readers who are only interested in certain parts of the material treated, are largely independent of each other. This arrangement has, of course, certain disadvantages such as repetitions and over-lappings.                                                                                                                                                                                                                                                                                                                                                                                                                                       |
-| [[https://archive.org/details/ElementaryDifferentialGeometry|{{:ga:elementary_differential_geometry-oneill.jpg?100}}]]                    | **Elementary Differential Geometry (1966)**\\ //Barrett O'Neill//\\ The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry. The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation.  |+| [[https://archive.org/details/ElementaryDifferentialGeometry|{{:ga:elementary_differential_geometry-oneill.jpg?100}}]] | **Elementary Differential Geometry (1966)**\\ //Barrett O'Neill//\\ The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry. The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation.  | 
 +| [[https://www.cambridge.org/core/books/topological-geometry/AAEBEBC695CF4A98242A74EA2C59E212|{{:ga:topological_geometry-porteous.jpg?100}}]] | **Topological Geometry, 2nd Edition (1981)**\\ //Ian R. Porteous//\\ The earlier chapters of this self-contained text provide a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place. In parallel with this is an account, also from first principles, of the elementary theory of topological spaces and of continuous and differentiable maps that leads up to the definitions of smooth manifolds and their tangent spaces and of Lie groups and Lie algebras. The calculus is presented as far as possible in basis free form to emphasize its geometrical flavor and its linear algebra content. In this second edition Dr. Porteous has taken the opportunity to add a chapter on triality which extends earlier work on the Spin groups in the chapter on Clifford algebras. The details include a number of important transitive group actions and a description of one of the exceptional Lie groups, the group G2.  |
  
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 ^                                                                                                                                                                                                         ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ^ ^                                                                                                                                                                                                         ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ^
-| [[https://www.amazon.com/Space-Time-Algebra-David-Hestenes/dp/3319184121|{{:ga:space-time_algebra-hestenes.jpg?650}}]]                                                                                  | **Space-Time Algebra, 2nd Ed (2015)**\\ //David Hestenes//\\ This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +| [[https://www.amazon.com/Space-Time-Algebra-David-Hestenes/dp/3319184121|{{:ga:space-time_algebra-hestenes.jpg?100}}]] | **Space-Time Algebra, 2nd Ed (2015)**\\ //David Hestenes//\\ This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | 
-| [[https://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148|{{:ga:new_foundations_for_classical_mechanics-hestenes.jpg?100}}]]                                          | **New Foundations for Classical Mechanics, 2nd Ed (1999)**\\ //David Hestenes//\\ This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels.                                                                                                                                                                                                                                                                                 | +| [[https://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148|{{:ga:new_foundations_for_classical_mechanics-hestenes.jpg?100}}]] | **New Foundations for Classical Mechanics, 2nd Ed (1999)**\\ //David Hestenes//\\ This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels.                                                                                                                                                                                                                                                                                 | 
-| [[https://www.amazon.com/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616|{{:ga:clifford_algebra_to_geometric_calculus-hestenes_sobczyk.jpg?100}}]]                                        | **Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (1984)**\\ //David Hestenes, Garret Sobczyk//\\ Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.                                                            | +| [[https://www.amazon.com/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616|{{:ga:clifford_algebra_to_geometric_calculus-hestenes_sobczyk.jpg?100}}]] | **Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (1984)**\\ //David Hestenes, Garret Sobczyk//\\ Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.                                                            | 
-| [[https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521480221|{{:ga:geometric_algebra_for_physicists-doran_lasenby.jpg?100}}]]                                                        | **Geometric Algebra for Physicists (2003)**\\ //Chris Doran, Anthony Lasenby//\\ This book is a complete guide to the current state of geometric algebra with early chapters providing a self-contained introduction. Topics range from new techniques for handling rotations in arbitrary dimensions, the links between rotations, bivectors, the structure of the Lie groups, non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +| [[https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521480221|{{:ga:geometric_algebra_for_physicists-doran_lasenby.jpg?100}}]] | **Geometric Algebra for Physicists (2003)**\\ //Chris Doran, Anthony Lasenby//\\ This book is a complete guide to the current state of geometric algebra with early chapters providing a self-contained introduction. Topics range from new techniques for handling rotations in arbitrary dimensions, the links between rotations, bivectors, the structure of the Lie groups, non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | 
-| [[https://www.amazon.com/Geometric-Algebra-Dover-Books-Mathematics/dp/0486801551|{{:ga:geometric_algebra-artin.jpg?100}}]]                                                                              | **Geometric Algebra (1957)**\\ //Emil Artin//\\ This concise classic presents advanced undergraduates and graduate students in mathematics with an overview of geometric algebra. The text originated with lecture notes from a New York University course taught by Emil Artin, one of the preeminent mathematicians of the twentieth century. The Bulletin of the American Mathematical Society praised Geometric Algebra upon its initial publication, noting that "mathematicians will find on many pages ample evidence of the author's ability to penetrate a subject and to present material in a particularly elegant manner."                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/Geometric-Algebra-Dover-Books-Mathematics/dp/0486801551|{{:ga:geometric_algebra-artin.jpg?100}}]] | **Geometric Algebra (1957)**\\ //Emil Artin//\\ This concise classic presents advanced undergraduates and graduate students in mathematics with an overview of geometric algebra. The text originated with lecture notes from a New York University course taught by Emil Artin, one of the preeminent mathematicians of the twentieth century. The Bulletin of the American Mathematical Society praised Geometric Algebra upon its initial publication, noting that "mathematicians will find on many pages ample evidence of the author's ability to penetrate a subject and to present material in a particularly elegant manner."                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | 
-| [[https://www.amazon.com/History-Vector-Analysis-Evolution-Mathematics/dp/0486679101|{{:ga:a_history_of_vector_analysis-crowe.jpg?100}}]]                                                               | **A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (1967)**\\ //Michael J. Crowe//\\ On October 16, 1843, Sir William Rowan Hamilton discovered quaternions and, on the very same day, presented his breakthrough to the Royal Irish Academy. Meanwhile, in a less dramatic style, a German high school teacher, Hermann Grassmann, was developing another vectorial system involving hypercomplex numbers comparable to quaternions. The creations of these two mathematicians led to other vectorial systems, most notably the system of vector analysis formulated by Josiah Willard Gibbs and Oliver Heaviside and now almost universally employed in mathematics, physics and engineering. Yet the Gibbs-Heaviside system won acceptance only after decades of debate and controversy in the latter half of the nineteenth century concerning which of the competing systems offered the greatest advantages for mathematical pedagogy and practice.                                                                                                                                                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/History-Vector-Analysis-Evolution-Mathematics/dp/0486679101|{{:ga:a_history_of_vector_analysis-crowe.jpg?100}}]] | **A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (1967)**\\ //Michael J. Crowe//\\ On October 16, 1843, Sir William Rowan Hamilton discovered quaternions and, on the very same day, presented his breakthrough to the Royal Irish Academy. Meanwhile, in a less dramatic style, a German high school teacher, Hermann Grassmann, was developing another vectorial system involving hypercomplex numbers comparable to quaternions. The creations of these two mathematicians led to other vectorial systems, most notably the system of vector analysis formulated by Josiah Willard Gibbs and Oliver Heaviside and now almost universally employed in mathematics, physics and engineering. Yet the Gibbs-Heaviside system won acceptance only after decades of debate and controversy in the latter half of the nineteenth century concerning which of the competing systems offered the greatest advantages for mathematical pedagogy and practice.                                                                                                                                                                                                                                                                                                                                                                                              | 
-| [[https://www.amazon.com/Theory-Spinors-Dover-Books-Mathematics/dp/0486640701|{{:ga:the_theory_of_spinors-cartan.jpg?100}}]]                                                                            | **The Theory of Spinors (1981)**\\ //Elie Cartan//\\ The French mathematician Elie Cartan (1869–1951) was one of the founders of the modern theory of Lie groups, a subject of central importance in mathematics and also one with many applications. In this volume, he describes the orthogonal groups, either with real or complex parameters including reflections, and also the related groups with indefinite metrics. He develops the theory of spinors (he discovered the general mathematical form of spinors in 1913) systematically by giving a purely geometrical definition of these mathematical entities; this geometrical origin makes it very easy to introduce spinors into Riemannian geometry, and particularly to apply the idea of parallel transport to these geometrical entities.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +| [[https://www.amazon.com/Theory-Spinors-Dover-Books-Mathematics/dp/0486640701|{{:ga:the_theory_of_spinors-cartan.jpg?100}}]] | **The Theory of Spinors (1981)**\\ //Elie Cartan//\\ The French mathematician Elie Cartan (1869–1951) was one of the founders of the modern theory of Lie groups, a subject of central importance in mathematics and also one with many applications. In this volume, he describes the orthogonal groups, either with real or complex parameters including reflections, and also the related groups with indefinite metrics. He develops the theory of spinors (he discovered the general mathematical form of spinors in 1913) systematically by giving a purely geometrical definition of these mathematical entities; this geometrical origin makes it very easy to introduce spinors into Riemannian geometry, and particularly to apply the idea of parallel transport to these geometrical entities.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | 
-| [[https://www.amazon.com/Past-Future-Gra%C3%9Fmanns-Bicentennial-Conference/dp/3034604041|{{:ga:hermann_grassmann_from_past_to_future.jpg?100}}]]                                                       | **From Past to Future: Graßmann's Work in Context: Graßmann Bicentennial Conference, September 2009 (2011)**\\ //Birkhauser//\\ On the occasion of the 200th anniversary of the birth of Hermann Graßmann (1809-1877), an interdisciplinary conference was held in Potsdam, Germany, and in Graßmann's hometown Szczecin, Poland. The idea of the conference was to present a multi-faceted picture of Graßmann, and to uncover the complexity of the factors that were responsible for his creativity. The conference demonstrated not only the very influential reception of his work at the turn of the 20th century, but also the unexpected modernity of his ideas, and their continuing development in the 21st century.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +| [[https://www.amazon.com/Past-Future-Gra%C3%9Fmanns-Bicentennial-Conference/dp/3034604041|{{:ga:hermann_grassmann_from_past_to_future.jpg?100}}]] | **From Past to Future: Graßmann's Work in Context: Graßmann Bicentennial Conference, September 2009 (2011)**\\ //Birkhauser//\\ On the occasion of the 200th anniversary of the birth of Hermann Graßmann (1809-1877), an interdisciplinary conference was held in Potsdam, Germany, and in Graßmann's hometown Szczecin, Poland. The idea of the conference was to present a multi-faceted picture of Graßmann, and to uncover the complexity of the factors that were responsible for his creativity. The conference demonstrated not only the very influential reception of his work at the turn of the 20th century, but also the unexpected modernity of his ideas, and their continuing development in the 21st century.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | 
-| [[https://www.amazon.com/Grassmann-Algebra-Foundations-Exploring-Mathematica/dp/1479197637|{{:ga:grassmann_algebra_volume_1_foundations-browne.jpg?100}}]]                                              | **Grassmann Algebra Volume 1: Foundations (2012)**\\ //John Browne//\\ Grassmann algebra extends vector algebra by introducing the exterior product to algebraicize the notion of linear dependence. With it, vectors may be extended to higher-grade entities: bivectors, trivectors, ... multivectors. The extensive exterior product also has a regressive dual: the regressive product. The pair behaves a little like the Boolean duals of union and intersection. By interpreting one of the elements of the vector space as an origin point, points can be defined, and the exterior product can extend points into higher-grade located entities from which lines, planes and multiplanes can be defined. Theorems of Projective Geometry are simply formulae involving these entities and the dual products. By introducing the (orthogonal) complement operation, the scalar product of vectors may be extended to the interior product of multivectors, which in this more general case may no longer result in a scalar. The notion of the magnitude of vectors is extended to the magnitude of multivectors: for example, the magnitude of the exterior product of two vectors (a bivector) is the area of the parallelogram formed by them. To develop these foundational concepts, we need only consider entities which are the sums of elements of the same grade. +| [[https://www.amazon.com/Grassmann-Algebra-Foundations-Exploring-Mathematica/dp/1479197637|{{:ga:grassmann_algebra_volume_1_foundations-browne.jpg?100}}]] | **Grassmann Algebra Volume 1: Foundations (2012)**\\ //John Browne//\\ Grassmann algebra extends vector algebra by introducing the exterior product to algebraicize the notion of linear dependence. With it, vectors may be extended to higher-grade entities: bivectors, trivectors, ... multivectors. The extensive exterior product also has a regressive dual: the regressive product. The pair behaves a little like the Boolean duals of union and intersection. By interpreting one of the elements of the vector space as an origin point, points can be defined, and the exterior product can extend points into higher-grade located entities from which lines, planes and multiplanes can be defined. Theorems of Projective Geometry are simply formulae involving these entities and the dual products. By introducing the (orthogonal) complement operation, the scalar product of vectors may be extended to the interior product of multivectors, which in this more general case may no longer result in a scalar. The notion of the magnitude of vectors is extended to the magnitude of multivectors: for example, the magnitude of the exterior product of two vectors (a bivector) is the area of the parallelogram formed by them. To develop these foundational concepts, we need only consider entities which are the sums of elements of the same grade. 
-| [[https://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932|{{:ga:linear_and_geometric_algebra-macdonald.jpg?100}}]]                                                                 | **Linear and Geometric Algebra (2011)**\\ //Alan Macdonald//\\ This textbook for the first undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics.Geometric algebra is an extension of linear algebra. It enhances the treatment of many linear algebra topics. And geometric algebra does much more.Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas. They provide a unified mathematical language for many areas of physics, computer science, and other fields.The book can be used for self study by those comfortable with the theorem/proof style of a mathematics text.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +| [[https://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932|{{:ga:linear_and_geometric_algebra-macdonald.jpg?100}}]] | **Linear and Geometric Algebra (2011)**\\ //Alan Macdonald//\\ This textbook for the first undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics.Geometric algebra is an extension of linear algebra. It enhances the treatment of many linear algebra topics. And geometric algebra does much more.Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas. They provide a unified mathematical language for many areas of physics, computer science, and other fields.The book can be used for self study by those comfortable with the theorem/proof style of a mathematics text.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | 
-| [[https://www.amazon.com/Vector-Geometric-Calculus-Alan-Macdonald/dp/1480132454|{{:ga:vector_and_geometric_calculus-macdonald.jpg?100}}]]                                                               | **Vector and Geometric Calculus (2012)**\\ //Alan Macdonald//\\ This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus. The book is a sequel to the text Linear and Geometric Algebra by the same author. Linear algebra and vector calculus have provided the basic vocabulary of mathematics in dimensions greater than one for the past one hundred years. Just as geometric algebra generalizes linear algebra in powerful ways, geometric calculus generalizes vector calculus in powerful ways. Traditional vector calculus topics are covered, as they must be, since readers will encounter them in other texts and out in the world. Differential geometry is used today in many disciplines. A final chapter is devoted to it.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +| [[https://www.amazon.com/Vector-Geometric-Calculus-Alan-Macdonald/dp/1480132454|{{:ga:vector_and_geometric_calculus-macdonald.jpg?100}}]] | **Vector and Geometric Calculus (2012)**\\ //Alan Macdonald//\\ This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus. The book is a sequel to the text Linear and Geometric Algebra by the same author. Linear algebra and vector calculus have provided the basic vocabulary of mathematics in dimensions greater than one for the past one hundred years. Just as geometric algebra generalizes linear algebra in powerful ways, geometric calculus generalizes vector calculus in powerful ways. Traditional vector calculus topics are covered, as they must be, since readers will encounter them in other texts and out in the world. Differential geometry is used today in many disciplines. A final chapter is devoted to it.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | 
-| [[https://www.amazon.com/Understanding-Geometric-Algebra-Hamilton-Grassmann/dp/1482259508|{{:ga:understanding_geometric_algebra-kanatani.jpg?100}}]]                                                    | **Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics (2015)**\\ //Kenichi Kanatani//\\ Introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton’s quaternion algebra, Grassmann’s outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres.                                                                                                                                                                           | +| [[https://www.amazon.com/Understanding-Geometric-Algebra-Hamilton-Grassmann/dp/1482259508|{{:ga:understanding_geometric_algebra-kanatani.jpg?100}}]] | **Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics (2015)**\\ //Kenichi Kanatani//\\ Introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton’s quaternion algebra, Grassmann’s outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres.                                                                                                                                                                           | 
-| [[https://www.amazon.com/Geometric-Algebra-Computer-Science-Revised/dp/0123749425|{{:ga:geometric_algebra_for_computer_science-dorst.jpg?100}}]]                                                        | **Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2007)**\\ //Leo Dorst,  Daniel Fontijne, Stephen Mann//\\ Until recently, all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex―often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs.                                                                                                                                                                                                                                                                                                                                                                                                                                             | +| [[https://www.amazon.com/Geometric-Algebra-Computer-Science-Revised/dp/0123749425|{{:ga:geometric_algebra_for_computer_science-dorst.jpg?100}}]] | **Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2007)**\\ //Leo Dorst,  Daniel Fontijne, Stephen Mann//\\ Until recently, all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex―often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs.                                                                                                                                                                                                                                                                                                                                                                                                                                             | 
-| [[https://www.amazon.com/New-Foundations-Mathematics-Geometric-Concept/dp/0817683844|{{:ga:new_foundations_in_mathematics-sobczyk.jpg?100}}]]                                                           | **New Foundations in Mathematics: The Geometric Concept of Number (2013)**\\ //Garret Sobczyk//\\ The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner.                                                                                                                                                                                                                                                                                                                                                                                                            | +| [[https://www.amazon.com/New-Foundations-Mathematics-Geometric-Concept/dp/0817683844|{{:ga:new_foundations_in_mathematics-sobczyk.jpg?100}}]] | **New Foundations in Mathematics: The Geometric Concept of Number (2013)**\\ //Garret Sobczyk//\\ The first book of its kind uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner.                                                                                                                                                                                                                                                                                                                                                                                                            
-| [[https://www.amazon.com/Clifford-Algebra-Computational-Tool-Physicists/dp/0195098242|{{:ga:clifford_algebra_a_computational_tool_for_physicists-snygg.jpg?100}}]]                                      | **Clifford Algebra: A Computational Tool for Physicists (1997)**\\ //John Snygg//\\ Clifford algebras have become an indispensable tool for physicists at the cutting edge of theoretical investigations. Applications in physics range from special relativity and the rotating top at one end of the spectrum, to general relativity and Dirac's equation for the electron at the other. Clifford algebras have also become a virtual necessity in some areas of physics, and their usefulness is expanding in other areas, such as algebraic manipulations involving Dirac matrices in quantum thermodynamics; Kaluza-Klein theories and dimensional renormalization theories; and the formation of superstring theories.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +| [[https://www.amazon.com/Clifford-Algebras-Classical-Cambridge-Mathematics/dp/0521551773|{{:ga:clifford_algebras_and_the_classical_groups-porteous.jpg?100}}]] | **Clifford Algebras and the Classical Groups (1995)**\\ //Ian R. Porteous//\\ This book reflects the growing interest in the theory of Clifford algebras and their applications. The author has reworked his previous book on this subject, Topological Geometry, and has expanded and added material. As in the previous version, the author includes an exhaustive treatment of all the generalizations of the classical groups, as well as an excellent exposition of the classification of the conjugation anti-involution of the Clifford algebras and their complexifications. Toward the end of the book, the author introduces ideas from the theory of Lie groups and Lie algebras.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | 
-| [[https://www.amazon.com/Approach-Differential-Geometry-Cliffords-Geometric/dp/0817682821|{{:ga:a_new_approach_to_differential_geometry_using_clifford_geometric_algebra-snygg.jpg?100}}]]              | **A New Approach to Differential Geometry using Clifford's Geometric Algebra (2012)**\\ //John Snygg//\\ Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +| [[https://www.amazon.com/Clifford-Algebras-Analysis-Advanced-Mathematics/dp/0849384818|{{:ga:clifford_algebras_in_analysis-ryan.jpg?100}}]] | **Clifford Algebras in Analysis and Related Topics (1995)**\\ //John Ryan//\\ Contains the most up-to-date and focused description of the applications of Clifford algebras in analysis, particularly classical harmonic analysis. Of particular interest is the contribution of Professor Alan McIntosh. He gives a detailed account of the links between Clifford algebras, monogenic and harmonic functions and the correspondence between monogenic functions and holomorphic functions of several complex variables under Fourier transforms. He describes the correspondence between algebras of singular integrals on Lipschitz surfaces and functional calculi of Dirac operators on these surfaces. He also discusses links with boundary value problems over Lipschitz domains. Other specific topics include Hardy spaces and compensated compactness in Euclidean space; applications to acoustic scattering and Galerkin estimates; scattering theory for orthogonal wavelets; applications of the conformal group and Vahalen matrices; Newmann type problems for the Dirac operator; plus much more.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
-| [[https://www.amazon.com/Clifford-Algebras-Spinors-Mathematical-Society/dp/0521005515|{{:ga:clifford_algebras_and_spinors-lounesto.jpg?100}}]]                                                          | **Clifford Algebras and Spinors, 2nd Ed (2001)**\\ //Pertti Lounesto//\\ The beginning chapters cover the basics: vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters, which will also interest physicists, include treatments of the quantum mechanics of the electron, electromagnetism and special relativity. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing among the Weyl, Majorana and Dirac spinors. Scalar products of spinors are categorized by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the algebraic side, Brauer/Wall groups and Witt rings are discussed, and on the analytic, Cauchy's integral formula is generalized to higher dimensions.                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +| [[https://www.amazon.com/Clifford-Algebra-Computational-Tool-Physicists/dp/0195098242|{{:ga:clifford_algebra_a_computational_tool_for_physicists-snygg.jpg?100}}]] | **Clifford Algebra: A Computational Tool for Physicists (1997)**\\ //John Snygg//\\ Clifford algebras have become an indispensable tool for physicists at the cutting edge of theoretical investigations. Applications in physics range from special relativity and the rotating top at one end of the spectrum, to general relativity and Dirac's equation for the electron at the other. Clifford algebras have also become a virtual necessity in some areas of physics, and their usefulness is expanding in other areas, such as algebraic manipulations involving Dirac matrices in quantum thermodynamics; Kaluza-Klein theories and dimensional renormalization theories; and the formation of superstring theories.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | 
-| [[https://www.amazon.com/Mathematics-Computer-Graphics-Undergraduate-Science/dp/1447162897|{{:ga:mathematics_for_computer_graphics-vince.jpg?100}}]]                                                    | **Mathematics for Computer Graphics, 4th Ed (2014)**\\ //John Vince//\\ Explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD, and other areas of computer graphics. Covering all the mathematical techniques required to resolve geometric problems and design computer programs for computer graphic applications, each chapter explores a specific mathematical topic prior to moving forward into the more advanced areas of matrix transforms, 3D curves and surface patches. Problem-solving techniques using vector analysis and geometric algebra are also discussed.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +| [[https://www.amazon.com/Approach-Differential-Geometry-Cliffords-Geometric/dp/0817682821|{{:ga:a_new_approach_to_differential_geometry_using_clifford_geometric_algebra-snygg.jpg?100}}]] | **A New Approach to Differential Geometry using Clifford's Geometric Algebra (2012)**\\ //John Snygg//\\ Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | 
-| [[https://www.amazon.com/Geometric-Algebra-Algebraic-Computer-Animation/dp/1848823789|{{:ga:geometric_algebra_an_algebraic_system_for_computer_games_and_animation-vince.jpg?100}}]]                    | **Geometric Algebra: An Algebraic System for Computer Games and Animation (2009)**\\ //John Vince//\\ Geometric algebra is still treated as an obscure branch of algebra and most books have been written by competent mathematicians in a very abstract style. This restricts the readership of such books especially by programmers working in computer graphics, who simply want guidance on algorithm design. Geometric algebra provides a unified algebraic system for solving a wide variety of geometric problems. John Vince reveals the beauty of this algebraic framework and communicates to the reader new and unusual mathematical concepts using colour illustrations, tabulations, and easy-to-follow algebraic proofs.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/Clifford-Algebras-Spinors-Mathematical-Society/dp/0521005515|{{:ga:clifford_algebras_and_spinors-lounesto.jpg?100}}]] | **Clifford Algebras and Spinors, 2nd Ed (2001)**\\ //Pertti Lounesto//\\ The beginning chapters cover the basics: vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters, which will also interest physicists, include treatments of the quantum mechanics of the electron, electromagnetism and special relativity. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing among the Weyl, Majorana and Dirac spinors. Scalar products of spinors are categorized by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the algebraic side, Brauer/Wall groups and Witt rings are discussed, and on the analytic, Cauchy's integral formula is generalized to higher dimensions.                                                                                                                                                                                                                                                                                                                                                                                                                                                           | 
-| [[https://www.amazon.com/Differential-Forms-Electromagnetics-Ismo-Lindell/dp/0471648019|{{:ga:differential_forms_in_electromagnetics-lindell.jpg?100}}]]                                                | **Differential Forms in Electromagnetics (2004)**\\ //Ismo V. Lindell//\\ An introduction to multivectors, dyadics, and differential forms for electrical engineers. While physicists have long applied differential forms to various areas of theoretical analysis, dyadic algebra is also the most natural language for expressing electromagnetic phenomena mathematically.  (...) Lindell simplifies the notation and adds memory aids in order to ease the reader's leap from Gibbsian analysis to differential forms, and provides the algebraic tools corresponding to the dyadics of Gibbsian analysis that have long been missing from the formalism. He introduces the reader to basic EM theory and wave equations for the electromagnetic two-forms, discusses the derivation of useful identities, and explains novel ways of treating problems in general linear (bi-anisotropic) media.                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/Mathematics-Computer-Graphics-Undergraduate-Science/dp/1447162897|{{:ga:mathematics_for_computer_graphics-vince.jpg?100}}]] | **Mathematics for Computer Graphics, 4th Ed (2014)**\\ //John Vince//\\ Explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD, and other areas of computer graphics. Covering all the mathematical techniques required to resolve geometric problems and design computer programs for computer graphic applications, each chapter explores a specific mathematical topic prior to moving forward into the more advanced areas of matrix transforms, 3D curves and surface patches. Problem-solving techniques using vector analysis and geometric algebra are also discussed.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | 
-| [[https://www.amazon.com/Understanding-Geometric-Algebra-Electromagnetic-Theory/dp/0470941634|{{:ga:understanding_geometric_algebra_for_electromagnetic_theory-arthur.jpg?100}}]]                       | **Understanding Geometric Algebra for Electromagnetic Theory (2011)**\\ //John W. Arthur//\\ This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison.                                                                                                                                                                                                                                                                                                                                                  | +| [[https://www.amazon.com/Geometric-Algebra-Algebraic-Computer-Animation/dp/1848823789|{{:ga:geometric_algebra_an_algebraic_system_for_computer_games_and_animation-vince.jpg?100}}]] | **Geometric Algebra: An Algebraic System for Computer Games and Animation (2009)**\\ //John Vince//\\ Geometric algebra is still treated as an obscure branch of algebra and most books have been written by competent mathematicians in a very abstract style. This restricts the readership of such books especially by programmers working in computer graphics, who simply want guidance on algorithm design. Geometric algebra provides a unified algebraic system for solving a wide variety of geometric problems. John Vince reveals the beauty of this algebraic framework and communicates to the reader new and unusual mathematical concepts using colour illustrations, tabulations, and easy-to-follow algebraic proofs.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | 
-| [[https://www.amazon.com/Geometric-Applications-Engineering-Geometry-Computing/dp/354089067X|{{:ga:geometric_algebra_with_applications_in_engineering-perwass.jpg?100}}]]                               | **Geometric Algebra with Applications in Engineering (2008)**\\ //Christian Perwass//\\ The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms. This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first part includes chapters on random variables in geometric algebra, linear estimation methods that incorporate the uncertainty of algebraic elements, and the representation of geometry in Euclidean, projective, conformal and conic space. The second part is dedicated to applications of geometric algebra, which include uncertain geometry and transformations, a generalized camera model, and pose estimation.                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/Differential-Forms-Electromagnetics-Ismo-Lindell/dp/0471648019|{{:ga:differential_forms_in_electromagnetics-lindell.jpg?100}}]] | **Differential Forms in Electromagnetics (2004)**\\ //Ismo V. Lindell//\\ An introduction to multivectors, dyadics, and differential forms for electrical engineers. While physicists have long applied differential forms to various areas of theoretical analysis, dyadic algebra is also the most natural language for expressing electromagnetic phenomena mathematically.  (...) Lindell simplifies the notation and adds memory aids in order to ease the reader's leap from Gibbsian analysis to differential forms, and provides the algebraic tools corresponding to the dyadics of Gibbsian analysis that have long been missing from the formalism. He introduces the reader to basic EM theory and wave equations for the electromagnetic two-forms, discusses the derivation of useful identities, and explains novel ways of treating problems in general linear (bi-anisotropic) media.                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | 
-| [[https://www.amazon.com/Foundations-Geometric-Algebra-Computing-Geometry/dp/3642317936|{{:ga:foundations_of_geometric_algebra_computing.jpg?100}}]]                                                    | **Foundations of Geometric Algebra Computing (2013)**\\ //Dietmar Hildenbrand//\\ The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +| [[https://www.amazon.com/Understanding-Geometric-Algebra-Electromagnetic-Theory/dp/0470941634|{{:ga:understanding_geometric_algebra_for_electromagnetic_theory-arthur.jpg?100}}]] | **Understanding Geometric Algebra for Electromagnetic Theory (2011)**\\ //John W. Arthur//\\ This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison.                                                                                                                                                                                                                                                                                                                                                  | 
-| [[https://www.amazon.com/Classical-Geometric-Algebra-Graduate-Mathematics/dp/0821820192|{{:ga:classical_groups_and_geometric_algebra-grove.jpg?100}}]]                                                  | **Classical Groups and Geometric Algebra (2001)**\\ //Larry C. Grove//\\  The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years, they have played a prominent role in the classification of the finite simple groups. This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +| [[https://www.amazon.com/Geometric-Applications-Engineering-Geometry-Computing/dp/354089067X|{{:ga:geometric_algebra_with_applications_in_engineering-perwass.jpg?100}}]] | **Geometric Algebra with Applications in Engineering (2008)**\\ //Christian Perwass//\\ The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms. This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first part includes chapters on random variables in geometric algebra, linear estimation methods that incorporate the uncertainty of algebraic elements, and the representation of geometry in Euclidean, projective, conformal and conic space. The second part is dedicated to applications of geometric algebra, which include uncertain geometry and transformations, a generalized camera model, and pose estimation.                                                                                                                                                                                                                                                              | 
-| [[https://www.amazon.com/Clifford-Geometric-Applications-Mathematics-Engineering/dp/0817638687|{{:ga:clifford_geometric_algebras-baylis.jpg?100}}]]                                                     | **Clifford (Geometric) Algebras with Applications in Physics, Mathematics, and Engineering (1999)**\\ //William Baylis//\\ The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas­ sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others.                                                                                                                                                                                                                                                                                                                                          | +| [[https://www.amazon.com/Foundations-Geometric-Algebra-Computing-Geometry/dp/3642317936|{{:ga:foundations_of_geometric_algebra_computing.jpg?100}}]] | **Foundations of Geometric Algebra Computing (2013)**\\ //Dietmar Hildenbrand//\\ The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
-| [[https://www.amazon.com/Guide-Geometric-Algebra-Practice-Dorst/dp/0857298100|{{:ga:guide_to_geometric_algebra_in_practice-dorst_lasenby.jpg?100}}]]                                                    | **Guide to Geometric Algebra in Practice (2011)**\\ //Leo Dorst, Joan Lasenby//\\ This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.                                                                                                                                                                                                                                                                                                                   | +| [[https://www.amazon.com/Classical-Geometric-Algebra-Graduate-Mathematics/dp/0821820192|{{:ga:classical_groups_and_geometric_algebra-grove.jpg?100}}]] | **Classical Groups and Geometric Algebra (2001)**\\ //Larry C. Grove//\\  The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years, they have played a prominent role in the classification of the finite simple groups. This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | 
-| [[https://www.amazon.com/Quaternions-Clifford-Algebras-Relativistic-Physics/dp/3764377909|{{:ga:quaternions_clifford_algebras_and_relativistic_physics-girard.jpg?100}}]]                               | **Quaternions, Clifford Algebras and Relativistic Physics (2007)**\\ //Patrick R. Girard//\\ The use of Clifford algebras in mathematical physics and engineering has grown rapidly in recent years. Whereas other developments have privileged a geometric approach, this book uses an algebraic approach that can be introduced as a tensor product of quaternion algebras and provides a unified calculus for much of physics. It proposes a pedagogical introduction to this new calculus, based on quaternions, with applications mainly in special relativity, classical electromagnetism, and general relativity.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +| [[https://www.amazon.com/Clifford-Geometric-Applications-Mathematics-Engineering/dp/0817638687|{{:ga:clifford_geometric_algebras-baylis.jpg?100}}]] | **Clifford (Geometric) Algebras with Applications in Physics, Mathematics, and Engineering (1999)**\\ //William Baylis//\\ The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas­ sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others.                                                                                                                                                                                                                                                                                                                                          | 
-| [[https://www.amazon.com/Geometric-Algebra-Applications-Physics-Sabbata/dp/1584887729|{{:ga:geometric_algebra_and_applications_to_physics-sabbata.jpg?100}}]]                                           | **Geometric Algebra and Applications to Physics (2006)**\\ //Venzo de Sabbata, Bidyut Kumar Datta//\\ Bringing geometric algebra to the mainstream of physics pedagogy, Geometric Algebra and Applications to Physics not only presents geometric algebra as a discipline within mathematical physics, but the book also shows how geometric algebra can be applied to numerous fundamental problems in physics, especially in experimental situations. This reference begins with several chapters that present the mathematical fundamentals of geometric algebra. It introduces the essential features of postulates and their underlying framework; bivectors, multivectors, and their operators; spinor and Lorentz rotations; and Clifford algebra. The book also extends some of these topics into three dimensions. Subsequent chapters apply these fundamentals to various common physical scenarios.                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +| [[https://www.amazon.com/Guide-Geometric-Algebra-Practice-Dorst/dp/0857298100|{{:ga:guide_to_geometric_algebra_in_practice-dorst_lasenby.jpg?100}}]] | **Guide to Geometric Algebra in Practice (2011)**\\ //Leo Dorst, Joan Lasenby//\\ This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.                                                                                                                                                                                                                                                                                                                   | 
-| [[https://www.amazon.com/Geometric-Computing-Clifford-Algebras-Gerald/dp/3540411984|{{:ga:geometric_computing_with_clifford_algebras-sommer.jpg?100}}]]                                                 | **Geometric Computing with Clifford Algebras (2001)**\\ //Gerald Sommer//\\ This monograph-like anthology introduces the concepts and framework of Clifford algebra. It provides a rich source of examples of how to work with this formalism. Clifford or geometric algebra shows strong unifying aspects and turned out in the 1960s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering. Recent work shows that Clifford algebra provides a universal and powerful algebraic framework for an elegant and coherent representation of various problems occurring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +| [[https://www.amazon.com/Quaternions-Clifford-Algebras-Relativistic-Physics/dp/3764377909|{{:ga:quaternions_clifford_algebras_and_relativistic_physics-girard.jpg?100}}]] | **Quaternions, Clifford Algebras and Relativistic Physics (2007)**\\ //Patrick R. Girard//\\ The use of Clifford algebras in mathematical physics and engineering has grown rapidly in recent years. Whereas other developments have privileged a geometric approach, this book uses an algebraic approach that can be introduced as a tensor product of quaternion algebras and provides a unified calculus for much of physics. It proposes a pedagogical introduction to this new calculus, based on quaternions, with applications mainly in special relativity, classical electromagnetism, and general relativity.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | 
-| [[https://www.amazon.com/Lectures-Clifford-Geometric-Algebras-Applications/dp/0817632573|{{:ga:lectures_on_clifford_geometric_algebras_and_applications-ablamowicz_sobczyk.jpg?100}}]]                  | **Lectures on Clifford (Geometric) Algebras and Applications (2004)**\\ //Rafal Ablamowicz, Garret Sobczyk//\\ The subject of Clifford (geometric) algebras offers a unified algebraic framework for the direct expression of the geometric concepts in algebra, geometry, and physics. This bird's-eye view of the discipline is presented by six of the world's leading experts in the field; it features an introductory chapter on Clifford algebras, followed by extensive explorations of their applications to physics, computer science, and differential geometry. The book is ideal for graduate students in mathematics, physics, and computer science; it is appropriate both for newcomers who have little prior knowledge of the field and professionals who wish to keep abreast of the latest applications.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +| [[https://www.amazon.com/Geometric-Algebra-Applications-Physics-Sabbata/dp/1584887729|{{:ga:geometric_algebra_and_applications_to_physics-sabbata.jpg?100}}]] | **Geometric Algebra and Applications to Physics (2006)**\\ //Venzo de Sabbata, Bidyut Kumar Datta//\\ Bringing geometric algebra to the mainstream of physics pedagogy, Geometric Algebra and Applications to Physics not only presents geometric algebra as a discipline within mathematical physics, but the book also shows how geometric algebra can be applied to numerous fundamental problems in physics, especially in experimental situations. This reference begins with several chapters that present the mathematical fundamentals of geometric algebra. It introduces the essential features of postulates and their underlying framework; bivectors, multivectors, and their operators; spinor and Lorentz rotations; and Clifford algebra. The book also extends some of these topics into three dimensions. Subsequent chapters apply these fundamentals to various common physical scenarios.                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | 
-| [[https://www.amazon.com/Geometric-Algebra-Applications-Science-Engineering/dp/0817641998|{{:ga:geometric_algebra_with_applications_in_science_and_engineering-bayro_sobczyk.jpg?100}}]]                | **Geometric Algebra with Applications in Science and Engineering (2001)**\\ //Eduardo Bayro-Corrochano, Garret Sobczyk//\\ The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer­ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in 1878 as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before. Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. The approach to Clifford algebra adopted in most of the ar­ticles here was pioneered in the 1960s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math­ematics and physics.                                                                                                                                                                                                                                                                                                                                                                                                               | +| [[https://www.amazon.com/Geometric-Computing-Clifford-Algebras-Gerald/dp/3540411984|{{:ga:geometric_computing_with_clifford_algebras-sommer.jpg?100}}]] | **Geometric Computing with Clifford Algebras (2001)**\\ //Gerald Sommer//\\ This monograph-like anthology introduces the concepts and framework of Clifford algebra. It provides a rich source of examples of how to work with this formalism. Clifford or geometric algebra shows strong unifying aspects and turned out in the 1960s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering. Recent work shows that Clifford algebra provides a universal and powerful algebraic framework for an elegant and coherent representation of various problems occurring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
-| [[https://www.amazon.com/Clifford-Algebras-Geometries-Application-Kinematics/dp/3658076178|{{:ga:clifford_algebras_geometric_modelling_and_chain_geometries-klawitter.jpg?100}}]]                       | **Clifford Algebras: Geometric Modelling and Chain Geometries with Application in Kinematics (2015)**\\ //Daniel Klawitter//\\ After revising known representations of the group of Euclidean displacements Daniel Klawitter gives a comprehensive introduction into Clifford algebras. The Clifford algebra calculus is used to construct new models that allow descriptions of the group of projective transformations and inversions with respect to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are examined. The author applies this theory and the developed methods to the homogeneous Clifford algebra model corresponding to Euclidean geometry. Moreover, kinematic mappings for special Cayley-Klein geometries are developed. These mappings allow a description of existing kinematic mappings in a unifying framework.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +| [[https://www.amazon.com/Lectures-Clifford-Geometric-Algebras-Applications/dp/0817632573|{{:ga:lectures_on_clifford_geometric_algebras_and_applications-ablamowicz_sobczyk.jpg?100}}]] | **Lectures on Clifford (Geometric) Algebras and Applications (2004)**\\ //Rafal Ablamowicz, Garret Sobczyk//\\ The subject of Clifford (geometric) algebras offers a unified algebraic framework for the direct expression of the geometric concepts in algebra, geometry, and physics. This bird's-eye view of the discipline is presented by six of the world's leading experts in the field; it features an introductory chapter on Clifford algebras, followed by extensive explorations of their applications to physics, computer science, and differential geometry. The book is ideal for graduate students in mathematics, physics, and computer science; it is appropriate both for newcomers who have little prior knowledge of the field and professionals who wish to keep abreast of the latest applications.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | 
-| [[https://www.amazon.com/Application-Geometric-Algebra-Electromagnetic-Scattering/dp/9811000883|{{:ga:application_of_geometric_algebra_to_electromagnetic_scattering-seagar.jpg?100}}]]                 | **Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique (2016)**\\ //Andrew Seagar//\\ This work presents the Clifford-Cauchy-Dirac (CCD) technique for solving problems involving the scattering of electromagnetic radiation from materials of all kinds. It allows anyone who is interested to master techniques that lead to simpler and more efficient solutions to problems of electromagnetic scattering than are currently in use. The technique is formulated in terms of the Cauchy kernel, single integrals, Clifford algebra and a whole-field approach. This is in contrast to many conventional techniques that are formulated in terms of Green's functions, double integrals, vector calculus and the combined field integral equation (CFIE).  Whereas these conventional techniques lead to an implementation using the method of moments (MoM), the CCD technique is implemented as alternating projections onto convex sets in a Banach space.                                                                                                                                                                                                                                                                                                                                                                    | +| [[https://www.amazon.com/Geometric-Algebra-Applications-Science-Engineering/dp/0817641998|{{:ga:geometric_algebra_with_applications_in_science_and_engineering-bayro_sobczyk.jpg?100}}]] | **Geometric Algebra with Applications in Science and Engineering (2001)**\\ //Eduardo Bayro-Corrochano, Garret Sobczyk//\\ The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer­ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in 1878 as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before. Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. The approach to Clifford algebra adopted in most of the ar­ticles here was pioneered in the 1960s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math­ematics and physics.                                                                                                                                                                                                                                                                                                                                                                                                               | 
-| [[https://www.amazon.com/Geometric-Algebra-Computing-Engineering-Computer/dp/1849961077|{{:ga:geometric_algebra_computing-bayro_scheuermann.jpg?100}}]]                                                 | **Geometric Algebra Computing: in Engineering and Computer Science (2010)**\\ //Eduardo Bayro-Corrochano, Gerik Scheuermann//\\ This useful text offers new insights and solutions for the development of theorems, algorithms and advanced methods for real-time applications across a range of disciplines. Its accessible style is enhanced by examples, figures and experimental analysis.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +| [[https://www.amazon.com/Clifford-Algebras-Geometries-Application-Kinematics/dp/3658076178|{{:ga:clifford_algebras_geometric_modelling_and_chain_geometries-klawitter.jpg?100}}]] | **Clifford Algebras: Geometric Modelling and Chain Geometries with Application in Kinematics (2015)**\\ //Daniel Klawitter//\\ After revising known representations of the group of Euclidean displacements Daniel Klawitter gives a comprehensive introduction into Clifford algebras. The Clifford algebra calculus is used to construct new models that allow descriptions of the group of projective transformations and inversions with respect to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are examined. The author applies this theory and the developed methods to the homogeneous Clifford algebra model corresponding to Euclidean geometry. Moreover, kinematic mappings for special Cayley-Klein geometries are developed. These mappings allow a description of existing kinematic mappings in a unifying framework.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | 
-| [[https://www.amazon.com/Handbook-Geometric-Computing-Applications-Neuralcomputing/dp/3540205950|{{:ga:handbook_of_geometric_computing-bayro.jpg?100}}]]                                                | **Handbook of Geometric Computing: Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics (2005)**\\ //Eduardo Bayro-Corrochano//\\ Many computer scientists, engineers, applied mathematicians, and physicists use geometry theory and geometric computing methods in the design of perception-action systems, intelligent autonomous systems, and man-machine interfaces. This handbook brings together the most recent advances in the application of geometric computing for building such systems, with contributions from leading experts in the important fields of neuroscience, neural networks, image processing, pattern recognition, computer vision, uncertainty in geometric computations, conformal computational geometry, computer graphics and visualization, medical imagery, geometry and robotics, and reaching and motion planning.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/Application-Geometric-Algebra-Electromagnetic-Scattering/dp/9811000883|{{:ga:application_of_geometric_algebra_to_electromagnetic_scattering-seagar.jpg?100}}]] | **Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique (2016)**\\ //Andrew Seagar//\\ This work presents the Clifford-Cauchy-Dirac (CCD) technique for solving problems involving the scattering of electromagnetic radiation from materials of all kinds. It allows anyone who is interested to master techniques that lead to simpler and more efficient solutions to problems of electromagnetic scattering than are currently in use. The technique is formulated in terms of the Cauchy kernel, single integrals, Clifford algebra and a whole-field approach. This is in contrast to many conventional techniques that are formulated in terms of Green's functions, double integrals, vector calculus and the combined field integral equation (CFIE).  Whereas these conventional techniques lead to an implementation using the method of moments (MoM), the CCD technique is implemented as alternating projections onto convex sets in a Banach space.                                                                                                                                                                                                                                                                                                                                                                    | 
-| [[https://www.amazon.com/Applications-Geometric-Algebra-Computer-Engineering/dp/1461266068|{{:ga:applications_of_geometric_algebra_in_computer_science_and_engineering-dorst_doran_lasenby.jpg?100}}]]  | **Applications of Geometric Algebra in Computer Science and Engineering (2002)**\\ //Leo Dorst, Chris Doran, Joan Lasenby//\\ Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant ideas are introduced in a self-contained manner and only a knowledge of linear algebra and calculus is assumed.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |+| [[https://www.amazon.com/Geometric-Algebra-Computing-Engineering-Computer/dp/1849961077|{{:ga:geometric_algebra_computing-bayro_scheuermann.jpg?100}}]] | **Geometric Algebra Computing: in Engineering and Computer Science (2010)**\\ //Eduardo Bayro-Corrochano, Gerik Scheuermann//\\ This useful text offers new insights and solutions for the development of theorems, algorithms and advanced methods for real-time applications across a range of disciplines. Its accessible style is enhanced by examples, figures and experimental analysis.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | 
 +| [[https://www.amazon.com/Handbook-Geometric-Computing-Applications-Neuralcomputing/dp/3540205950|{{:ga:handbook_of_geometric_computing-bayro.jpg?100}}]] | **Handbook of Geometric Computing: Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics (2005)**\\ //Eduardo Bayro-Corrochano//\\ Many computer scientists, engineers, applied mathematicians, and physicists use geometry theory and geometric computing methods in the design of perception-action systems, intelligent autonomous systems, and man-machine interfaces. This handbook brings together the most recent advances in the application of geometric computing for building such systems, with contributions from leading experts in the important fields of neuroscience, neural networks, image processing, pattern recognition, computer vision, uncertainty in geometric computations, conformal computational geometry, computer graphics and visualization, medical imagery, geometry and robotics, and reaching and motion planning.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | 
 +| [[https://www.amazon.com/Applications-Geometric-Algebra-Computer-Engineering/dp/1461266068|{{:ga:applications_of_geometric_algebra_in_computer_science_and_engineering-dorst_doran_lasenby.jpg?100}}]] | **Applications of Geometric Algebra in Computer Science and Engineering (2002)**\\ //Leo Dorst, Chris Doran, Joan Lasenby//\\ Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant ideas are introduced in a self-contained manner and only a knowledge of linear algebra and calculus is assumed.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
 | [[https://www.amazon.com/Operator-Calculus-Graphs-Applications-Computer/dp/1848168764|{{:ga:operator_calculus_on_graphs-schott.jpg?100}}]]                                                              | **Operator Calculus On Graphs: Theory and Applications in Computer Science (2012)**\\ //Rene Schott, G. Stacey Staples//\\ This pioneering book presents a study of the interrelationships among operator calculus, graph theory, and quantum probability in a unified manner, with significant emphasis on symbolic computations and an eye toward applications in computer science. Presented in this book are new methods, built on the algebraic framework of Clifford algebras, for tackling important real world problems related, but not limited to, wireless communications, neural networks, electrical circuits, transportation, and the world wide web.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | | [[https://www.amazon.com/Operator-Calculus-Graphs-Applications-Computer/dp/1848168764|{{:ga:operator_calculus_on_graphs-schott.jpg?100}}]]                                                              | **Operator Calculus On Graphs: Theory and Applications in Computer Science (2012)**\\ //Rene Schott, G. Stacey Staples//\\ This pioneering book presents a study of the interrelationships among operator calculus, graph theory, and quantum probability in a unified manner, with significant emphasis on symbolic computations and an eye toward applications in computer science. Presented in this book are new methods, built on the algebraic framework of Clifford algebras, for tackling important real world problems related, but not limited to, wireless communications, neural networks, electrical circuits, transportation, and the world wide web.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
-| [[https://www.amazon.com/Multivectors-Clifford-Algebra-Electrodynamics-Jancewicz/dp/9971502909|{{:ga:multivectors_and_clifford_algebra_in_electrodynamics-jancewicz.jpg?100}}]]                         | **Multivectors And Clifford Algebra In Electrodynamics (1989)**\\ //Bernard Jancewicz//\\ Clifford algebras are assuming now an increasing role in theoretical physics. Some of them predominantly larger ones are used in elementary particle theory, especially for a unification of the fundamental interactions. The smaller ones are promoted in more classical domains. This book is intended to demonstrate usefulness of Clifford algebras in classical electrodynamics. Written with a pedagogical aim, it begins with an introductory chapter devoted to multivectors and Clifford algebra for the three-dimensional space. In a later chapter modifications are presented necessary for higher dimension and for the pseudoeuclidean metric of the Minkowski space. Among other advantages one is worth mentioning: Due to a bivectorial description of the magnetic field a notion of force surfaces naturally emerges, which reveals an intimate link between the magnetic field and the electric currents as its sources.                                                                                                                                                                                                                                                                                                                                             | +| [[https://www.amazon.com/Multivectors-Clifford-Algebra-Electrodynamics-Jancewicz/dp/9971502909|{{:ga:multivectors_and_clifford_algebra_in_electrodynamics-jancewicz.jpg?100}}]] | **Multivectors And Clifford Algebra In Electrodynamics (1989)**\\ //Bernard Jancewicz//\\ Clifford algebras are assuming now an increasing role in theoretical physics. Some of them predominantly larger ones are used in elementary particle theory, especially for a unification of the fundamental interactions. The smaller ones are promoted in more classical domains. This book is intended to demonstrate usefulness of Clifford algebras in classical electrodynamics. Written with a pedagogical aim, it begins with an introductory chapter devoted to multivectors and Clifford algebra for the three-dimensional space. In a later chapter modifications are presented necessary for higher dimension and for the pseudoeuclidean metric of the Minkowski space. Among other advantages one is worth mentioning: Due to a bivectorial description of the magnetic field a notion of force surfaces naturally emerges, which reveals an intimate link between the magnetic field and the electric currents as its sources.                                                                                                                                                                                                                                                                                                                                             | 
-| [[https://www.amazon.com/Clifford-Algebras-Applications-Mathematical-Physics/dp/0817641823|{{:ga:clifford_algebras_and_their_applications_in_mathematical_physics_vol1-ablamowicz.jpg?100}}]]           | **Clifford Algebras and Their Applications in Mathematical Physics, Vol.1: Algebra and Physics (2000)**\\ //Rafal Ablamowicz, Bertfried Fauser//\\ The first part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras and their applications in physics. Algebraic geometry, cohomology, non-communicative spaces, q-deformations and the related quantum groups, and projective geometry provide the basis for algebraic topics covered. Physical applications and extensions of physical theories such as the theory of quaternionic spin, a projective theory of hadron transformation laws, and electron scattering are also presented, showing the broad applicability of Clifford geometric algebras in solving physical problems.                                                                                                                                                                                                                                                                                                                                                                                                                                | +| [[https://www.amazon.com/Clifford-Algebras-Applications-Mathematical-Physics/dp/0817641823|{{:ga:clifford_algebras_and_their_applications_in_mathematical_physics_vol1-ablamowicz.jpg?100}}]] | **Clifford Algebras and Their Applications in Mathematical Physics, Vol.1: Algebra and Physics (2000)**\\ //Rafal Ablamowicz, Bertfried Fauser//\\ The first part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras and their applications in physics. Algebraic geometry, cohomology, non-communicative spaces, q-deformations and the related quantum groups, and projective geometry provide the basis for algebraic topics covered. Physical applications and extensions of physical theories such as the theory of quaternionic spin, a projective theory of hadron transformation laws, and electron scattering are also presented, showing the broad applicability of Clifford geometric algebras in solving physical problems.                                                                                                                                                                                                                                                                                                                                                                                                                                | 
-| [[https://www.amazon.com/Clifford-Algebras-Applications-Mathematical-Physics/dp/0817641831|{{:ga:clifford_algebras_and_their_applications_in_mathematical_physics_vol2-ablamowicz.jpg?100}}]]           | **Clifford Algebras and Their Applications in Mathematical Physics, Vol. 2: Clifford Analysis (2000)**\\ //John Ryan, Wolfgang Sproessig//\\ The second part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras. from applications such as complex-distance potential theory, supersymmetry, and fluid dynamics to Fourier analysis, the study of boundary value problems, and applications, to mathematical physics and Schwarzian derivatives in Euclidean space. Among the mathematical topics examined are generalized Dirac operators, holonomy groups, monogenic and hypermonogenic functions and their derivatives, quaternionic Beltrami equations, Fourier theory under Mobius transformations, Cauchy-Reimann operators, and Cauchy type integrals.                                                                                                                                                                                                                                                                                                                                                                                                          | +| [[https://www.amazon.com/Clifford-Algebras-Applications-Mathematical-Physics/dp/0817641831|{{:ga:clifford_algebras_and_their_applications_in_mathematical_physics_vol2-ablamowicz.jpg?100}}]] | **Clifford Algebras and Their Applications in Mathematical Physics, Vol. 2: Clifford Analysis (2000)**\\ //John Ryan, Wolfgang Sproessig//\\ The second part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras. from applications such as complex-distance potential theory, supersymmetry, and fluid dynamics to Fourier analysis, the study of boundary value problems, and applications, to mathematical physics and Schwarzian derivatives in Euclidean space. Among the mathematical topics examined are generalized Dirac operators, holonomy groups, monogenic and hypermonogenic functions and their derivatives, quaternionic Beltrami equations, Fourier theory under Mobius transformations, Cauchy-Reimann operators, and Cauchy type integrals.                                                                                                                                                                                                                                                                                                                                                                                                          | 
-| [[https://www.amazon.com/Clifford-Algebras-Spinor-Structures-Applications/dp/9048145252|{{:ga:clifford_algebras_and_spinor_structures-ablamowicz.jpg?100}}]]                                            | **Clifford Algebras and Spinor Structures (1995)**\\ //Rafal Ablamowicz, Pertti Lounesto//\\ This volume is dedicated to the memory of Albert Crumeyrolle, who died on June 17, 1992. In organizing the volume we gave priority to: articles summarizing Crumeyrolle's own work in differential geometry, general relativity and spinors, articles which give the reader an idea of the depth and breadth of Crumeyrolle's research interests and influence in the field, articles of high scientific quality which would be of general interest. In each of the areas to which Crumeyrolle made significant contribution - Clifford and exterior algebras, Weyl and pure spinors, spin structures on manifolds, principle of triality, conformal geometry - there has been substantial progress.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +| [[https://www.amazon.com/Clifford-Algebras-Spinor-Structures-Applications/dp/9048145252|{{:ga:clifford_algebras_and_spinor_structures-ablamowicz.jpg?100}}]] | **Clifford Algebras and Spinor Structures (1995)**\\ //Rafal Ablamowicz, Pertti Lounesto//\\ This volume is dedicated to the memory of Albert Crumeyrolle, who died on June 17, 1992. In organizing the volume we gave priority to: articles summarizing Crumeyrolle's own work in differential geometry, general relativity and spinors, articles which give the reader an idea of the depth and breadth of Crumeyrolle's research interests and influence in the field, articles of high scientific quality which would be of general interest. In each of the areas to which Crumeyrolle made significant contribution - Clifford and exterior algebras, Weyl and pure spinors, spin structures on manifolds, principle of triality, conformal geometry - there has been substantial progress.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | 
-| [[https://www.amazon.com/Quaternionic-Clifford-Calculus-Physicists-Engineers/dp/0471962007|{{:ga:quaternionic_and_clifford_calculus_for_physicists_and_engineers-gurlebeck.jpg?100}}]]                  | **Quaternionic and Clifford Calculus for Physicists and Engineers (1998)**\\ //Klaus Gürlebeck, Wolfgang Sprössig//\\ Quarternionic calculus covers a branch of mathematics which uses computational techniques to help solve problems from a wide variety of physical systems which are mathematically modelled in 3, 4 or more dimensions. Examples of the application areas include thermodynamics, hydrodynamics, geophysics and structural mechanics. Focusing on the Clifford algebra approach the authors have drawn together the research into quarternionic calculus to provide the non-expert or research student with an accessible introduction to the subject.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +| [[https://www.amazon.com/Quaternionic-Clifford-Calculus-Physicists-Engineers/dp/0471962007|{{:ga:quaternionic_and_clifford_calculus_for_physicists_and_engineers-gurlebeck.jpg?100}}]] | **Quaternionic and Clifford Calculus for Physicists and Engineers (1998)**\\ //Klaus Gürlebeck, Wolfgang Sprössig//\\ Quarternionic calculus covers a branch of mathematics which uses computational techniques to help solve problems from a wide variety of physical systems which are mathematically modelled in 3, 4 or more dimensions. Examples of the application areas include thermodynamics, hydrodynamics, geophysics and structural mechanics. Focusing on the Clifford algebra approach the authors have drawn together the research into quarternionic calculus to provide the non-expert or research student with an accessible introduction to the subject.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | 
-| [[https://www.amazon.com/Clifford-Algebras-Numeric-Symbolic-Computations/dp/1461581591|{{:ga:clifford_algebras_with_numeric_and_symbolic_computations-ablamowicz.jpg?100}}]]                            | **Clifford Algebras with Numeric and Symbolic Computations (1996)**\\ //Rafal Ablamowicz, Joseph Parra, Pertti Lounesto//\\ This edited survey book consists of 20 chapters showing application of Clifford algebra in quantum mechanics, field theory, spinor calculations, projective geometry, Hypercomplex algebra, function theory and crystallography. Many examples of computations performed with a variety of readily available software programs are presented in detail.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +| [[https://www.amazon.com/Clifford-Algebras-Numeric-Symbolic-Computations/dp/1461581591|{{:ga:clifford_algebras_with_numeric_and_symbolic_computations-ablamowicz.jpg?100}}]] | **Clifford Algebras with Numeric and Symbolic Computations (1996)**\\ //Rafal Ablamowicz, Joseph Parra, Pertti Lounesto//\\ This edited survey book consists of 20 chapters showing application of Clifford algebra in quantum mechanics, field theory, spinor calculations, projective geometry, Hypercomplex algebra, function theory and crystallography. Many examples of computations performed with a variety of readily available software programs are presented in detail.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | 
-| [[https://www.amazon.com/Quadratic-Mappings-Clifford-Algebras-Helmstetter/dp/3764386053|{{:ga:quadratic_mappings_and_clifford_algebras-helmstetter.jpg?100}}]]                                          | **Quadratic Mappings and Clifford Algebras (2008)**\\ //Jacques Helmstetter, Artibano Micali//\\ After general properties of quadratic mappings over rings, the authors more intensely study quadratic forms, and especially their Clifford algebras. To this purpose they review the required part of commutative algebra, and they present a significant part of the theory of graded Azumaya algebras. Interior multiplications and deformations of Clifford algebras are treated with the most efficient methods.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +| [[https://www.amazon.com/Quadratic-Mappings-Clifford-Algebras-Helmstetter/dp/3764386053|{{:ga:quadratic_mappings_and_clifford_algebras-helmstetter.jpg?100}}]] | **Quadratic Mappings and Clifford Algebras (2008)**\\ //Jacques Helmstetter, Artibano Micali//\\ After general properties of quadratic mappings over rings, the authors more intensely study quadratic forms, and especially their Clifford algebras. To this purpose they review the required part of commutative algebra, and they present a significant part of the theory of graded Azumaya algebras. Interior multiplications and deformations of Clifford algebras are treated with the most efficient methods.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
-| [[https://www.amazon.com/Algebraic-Theory-Spinors-Clifford-Algebras/dp/3540570632|{{:ga:the_algebraic_theory_of_spinors_and_clifford_algebras-chevalley.jpg?100}}]]                                     | **The Algebraic Theory of Spinors and Clifford Algebras (1997)**\\ //Claude Chevalley, Pierre Cartier, Catherine Chevalley//\\ This volume is Vol. 2 of a projected series devoted to the mathematical and philosophical works of the late Claude Chevalley. It covers the main contributions by the author to the theory of spinors. Since its appearance in 1954, "The Algebraic Theory of Spinors" has been a very sought after reference. It presents the whole story of one subject in a concise and especially clear manner. The reprint of the book is supplemented by a series of lectures on Clifford Algebras given by the author in Japan at about the same time. Also included is a postface by J. P. Bourguignon describing the many uses of spinors in differential geometry developed by mathematical physicists from the 1970s to the present day. After its appearance the book was reviewed at length by Jean Dieudonné. His insightful criticism of the book is also made available to the reader in this volume.                                                                                                                                                                                                                                                                                                                                                | +| [[https://www.amazon.com/Algebraic-Theory-Spinors-Clifford-Algebras/dp/3540570632|{{:ga:the_algebraic_theory_of_spinors_and_clifford_algebras-chevalley.jpg?100}}]] | **The Algebraic Theory of Spinors and Clifford Algebras (1997)**\\ //Claude Chevalley, Pierre Cartier, Catherine Chevalley//\\ This volume is Vol. 2 of a projected series devoted to the mathematical and philosophical works of the late Claude Chevalley. It covers the main contributions by the author to the theory of spinors. Since its appearance in 1954, "The Algebraic Theory of Spinors" has been a very sought after reference. It presents the whole story of one subject in a concise and especially clear manner. The reprint of the book is supplemented by a series of lectures on Clifford Algebras given by the author in Japan at about the same time. Also included is a postface by J. P. Bourguignon describing the many uses of spinors in differential geometry developed by mathematical physicists from the 1970s to the present day. After its appearance the book was reviewed at length by Jean Dieudonné. His insightful criticism of the book is also made available to the reader in this volume.                                                                                                                                                                                                                                                                                                                                                | 
-| [[https://www.amazon.com/Faces-Maxwell-Dirac-Einstein-Equations/dp/3319276360|{{:ga:the_many_faces_of_maxwell_dirac_and_einstein_equations-rodrigues.jpg?100}}]]                                        | **The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach (2016)**\\ //Waldyr A. Rodrigues Jr, Edmundo Capelas de Oliveira//\\ This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +| [[https://www.amazon.com/Faces-Maxwell-Dirac-Einstein-Equations/dp/3319276360|{{:ga:the_many_faces_of_maxwell_dirac_and_einstein_equations-rodrigues.jpg?100}}]] | **The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach (2016)**\\ //Waldyr A. Rodrigues Jr, Edmundo Capelas de Oliveira//\\ This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | 
-| [[https://www.amazon.com/Clifford-Ergebnisse-Mathematik-Grenzgebiete-Mathematics/dp/364236215X|{{:ga:clifford_algebras_and_lie_theory-meinrenken.jpg?100}}]]                                            | **Clifford Algebras and Lie Theory (2013)**\\ //Eckhard Meinrenken//\\ This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré-Birkhoff-Witt theorem. This is followed by discussions of Weil algebras, Chern-Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.                                                                                                                     | +| [[https://www.amazon.com/Clifford-Ergebnisse-Mathematik-Grenzgebiete-Mathematics/dp/364236215X|{{:ga:clifford_algebras_and_lie_theory-meinrenken.jpg?100}}]] | **Clifford Algebras and Lie Theory (2013)**\\ //Eckhard Meinrenken//\\ This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré-Birkhoff-Witt theorem. This is followed by discussions of Weil algebras, Chern-Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.                                                                                                                     | 
-| [[https://www.amazon.com/Introduction-Clifford-Algebras-Spinors/dp/0198782926|{{:ga:an_introduction_to_clifford_algebras_and_spinors-vaz.jpg?100}}]]                                                    | **An Introduction to Clifford Algebras and Spinors (2016)**\\ // Jayme Vaz, Roldao da Rocha//\\ This text explores how Clifford algebras and spinors have been sparking a collaboration and bridging a gap between Physics and Mathematics. This collaboration has been the consequence of a growing awareness of the importance of algebraic and geometric properties in many physical phenomena, and of the discovery of common ground through various touch points: relating Clifford algebras and the arising geometry to so-called spinors, and to their three definitions (both from the mathematical and physical viewpoint). The main point of contact are the representations of Clifford algebras and the periodicity theorems. Clifford algebras also constitute a highly intuitive formalism, having an intimate relationship to quantum field theory. The text strives to seamlessly combine these various viewpoints and is devoted to a wider audience of both physicists and mathematicians.                                                                                                                                                                                                                                                                                                                                                                        | +| [[https://www.amazon.com/Introduction-Clifford-Algebras-Spinors/dp/0198782926|{{:ga:an_introduction_to_clifford_algebras_and_spinors-vaz.jpg?100}}]] | **An Introduction to Clifford Algebras and Spinors (2016)**\\ // Jayme Vaz, Roldao da Rocha//\\ This text explores how Clifford algebras and spinors have been sparking a collaboration and bridging a gap between Physics and Mathematics. This collaboration has been the consequence of a growing awareness of the importance of algebraic and geometric properties in many physical phenomena, and of the discovery of common ground through various touch points: relating Clifford algebras and the arising geometry to so-called spinors, and to their three definitions (both from the mathematical and physical viewpoint). The main point of contact are the representations of Clifford algebras and the periodicity theorems. Clifford algebras also constitute a highly intuitive formalism, having an intimate relationship to quantum field theory. The text strives to seamlessly combine these various viewpoints and is devoted to a wider audience of both physicists and mathematicians.                                                                                                                                                                                                                                                                                                                                                                        | 
-| [[https://www.amazon.com/Clifford-Algebras-Introduction-Mathematical-Society/dp/1107096383|{{:ga:clifford_algebras_an_introduction-garling.jpg?100}}]]                                                  | **Clifford Algebras: An Introduction (2011)**\\ //D. J. H. Garling//\\ Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions. They are famously used in proofs of the Atiyah-Singer index theorem, to provide double covers (spin groups) of the classical groups and to generalize the Hilbert transform. They also have their place in physics, setting the scene for Maxwell's equations in electromagnetic theory, for the spin of elementary particles and for the Dirac equation. This straightforward introduction to Clifford algebras makes the necessary algebraic background - including multilinear algebra, quadratic spaces and finite-dimensional real algebras - easily accessible to research students and final-year undergraduates. The author also introduces many applications in mathematics and physics, equipping the reader with Clifford algebras as a working tool in a variety of contexts.                                                                                                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/Modern-Trends-Hypercomplex-Analysis-Mathematics/dp/3319425285|{{:ga:modern_trends_in_hypercomplex_analysis-birkhauser.jpg?100}}]] | **Modern Trends in Hypercomplex Analysis (2016)**\\ //  Swanhild Bernstein, Uwe Kähler, Irene Sabadini, Franciscus Sommen (Editors)//\\ This book contains a selection of papers presented at the session "Quaternionic and Clifford Analysis" at the 10th ISAAC Congress held in Macau in August 2015. The covered topics represent the state-of-the-art as well as new trends in hypercomplex analysis and its applications.                                                                                                                                                                                                                                                                                                                                                                        | 
-| [[https://www.amazon.com/Standard-Quantum-Physics-Clifford-Algebra/dp/9814719862|{{:ga:standard_model_of_quantum_physics_in_clifford_algebra-daviau.jpg?100}}]]                                         | **The Standard Model of Quantum Physics in Clifford Algebra (2015)**\\ //Claude Daviau, Jacques Bertrand//\\ We extend to gravitation our previous study of a quantum wave for all particles and antiparticles of each generation (electron + neutrino + u and d quarks for instance). This wave equation is form invariant under Cl*_3, then relativistic invariant. It is gauge invariant under the gauge group of the standard model, with a mass term: this was impossible before, and the consequence was an impossibility to link gauge interactions and gravitation.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +| [[https://www.amazon.com/Clifford-Algebras-Introduction-Mathematical-Society/dp/1107096383|{{:ga:clifford_algebras_an_introduction-garling.jpg?100}}]] | **Clifford Algebras: An Introduction (2011)**\\ //D. J. H. Garling//\\ Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions. They are famously used in proofs of the Atiyah-Singer index theorem, to provide double covers (spin groups) of the classical groups and to generalize the Hilbert transform. They also have their place in physics, setting the scene for Maxwell's equations in electromagnetic theory, for the spin of elementary particles and for the Dirac equation. This straightforward introduction to Clifford algebras makes the necessary algebraic background - including multilinear algebra, quadratic spaces and finite-dimensional real algebras - easily accessible to research students and final-year undergraduates. The author also introduces many applications in mathematics and physics, equipping the reader with Clifford algebras as a working tool in a variety of contexts.                                                                                                                                                                                                                                                                                                                                              | 
-| [[https://www.amazon.com/lunification-mathematiques-algebresgeometriques-algebrique-informatique/dp/2746238381|{{:ga:lunification_des_mathematiques-parrochia.jpg?100}}]]                               | **L'unification des mathématiques: algèbres géométriques, géométrie algébrique et philosophie de Langlands (2012)**\\ //Daniel Parrochia, Artibano Micali, Pierre Anglès//\\ La pensée mathématique offre un panorama impressionnant de recherches dans les multiples directions dessinées par les réorganisations successives que la matière a connues. Cet ouvrage porte un éclairage philosophique et historique sur certains développements qui donne un sens aux transformations subies par la pensée mathématique au cours du temps pour actualiser le portrait déjà ancien de "l'unité des mathématiques". Deux mouvements symétriques d'unification se sont produits en mathématiques. Le premier est l'aboutissement du long chemin qui, depuis les Grecs, a tendu à résoudre l'opposition de la géométrie et de l'arithmétique, puis de la géométrie et de l'algèbre. Le second mode d'unification date de la fin des années 1960. Via la géométrie algébrique, il tend à reconstruire l'ensemble des mathématiques sur la base des correspondances de Langlands, lesquelles résorbent intégralement l'opposition de l'algèbre et de l'analyse, et constituent un fabuleux dictionnaire pour la physique de demain.                                                                                                                                                       | +| [[https://www.amazon.com/Standard-Quantum-Physics-Clifford-Algebra/dp/9814719862|{{:ga:standard_model_of_quantum_physics_in_clifford_algebra-daviau.jpg?100}}]] | **The Standard Model of Quantum Physics in Clifford Algebra (2015)**\\ //Claude Daviau, Jacques Bertrand//\\ We extend to gravitation our previous study of a quantum wave for all particles and antiparticles of each generation (electron + neutrino + u and d quarks for instance). This wave equation is form invariant under Cl*_3, then relativistic invariant. It is gauge invariant under the gauge group of the standard model, with a mass term: this was impossible before, and the consequence was an impossibility to link gauge interactions and gravitation.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | 
-| [[http://www.lulu.com/shop/sergei-winitzki/linear-algebra-via-exterior-products/paperback/product-6214034.html|{{:ga:linear_algebra_via_exterior_products-winitzki.jpg?100}}]]                          | ** Linear Algebra via Exterior Products (2010)**\\ //Sergei Winitzki//\\ This is a pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. The reader should be already exposed to the array-based formalism of vector and matrix calculations. This book makes extensive use of the exterior (anti-commutative, "wedge") product of vectors. The coordinate-free formalism and the exterior product, while somewhat more abstract, provide a deeper understanding of the classical results in linear algebra. Without cumbersome matrix calculations, this text derives the standard properties of determinants, the Pythagorean formula for multidimensional volumes, the formulas of Jacobi and Liouville, the Cayley-Hamilton theorem, the Jordan canonical form, the properties of Pfaffians, as well as some generalizations of these results.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |+| [[https://www.amazon.com/lunification-mathematiques-algebresgeometriques-algebrique-informatique/dp/2746238381|{{:ga:lunification_des_mathematiques-parrochia.jpg?100}}]] | **L'unification des mathématiques: algèbres géométriques, géométrie algébrique et philosophie de Langlands (2012)**\\ //Daniel Parrochia, Artibano Micali, Pierre Anglès//\\ La pensée mathématique offre un panorama impressionnant de recherches dans les multiples directions dessinées par les réorganisations successives que la matière a connues. Cet ouvrage porte un éclairage philosophique et historique sur certains développements qui donne un sens aux transformations subies par la pensée mathématique au cours du temps pour actualiser le portrait déjà ancien de "l'unité des mathématiques". Deux mouvements symétriques d'unification se sont produits en mathématiques. Le premier est l'aboutissement du long chemin qui, depuis les Grecs, a tendu à résoudre l'opposition de la géométrie et de l'arithmétique, puis de la géométrie et de l'algèbre. Le second mode d'unification date de la fin des années 1960. Via la géométrie algébrique, il tend à reconstruire l'ensemble des mathématiques sur la base des correspondances de Langlands, lesquelles résorbent intégralement l'opposition de l'algèbre et de l'analyse, et constituent un fabuleux dictionnaire pour la physique de demain.                                                                                                                                                       | 
 +| [[http://www.lulu.com/shop/sergei-winitzki/linear-algebra-via-exterior-products/paperback/product-6214034.html|{{:ga:linear_algebra_via_exterior_products-winitzki.jpg?100}}]] | ** Linear Algebra via Exterior Products (2010)**\\ //Sergei Winitzki//\\ This is a pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. The reader should be already exposed to the array-based formalism of vector and matrix calculations. This book makes extensive use of the exterior (anti-commutative, "wedge") product of vectors. The coordinate-free formalism and the exterior product, while somewhat more abstract, provide a deeper understanding of the classical results in linear algebra. Without cumbersome matrix calculations, this text derives the standard properties of determinants, the Pythagorean formula for multidimensional volumes, the formulas of Jacobi and Liouville, the Cayley-Hamilton theorem, the Jordan canonical form, the properties of Pfaffians, as well as some generalizations of these results.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
 | [[https://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311|{{:ga:the_road_to_reality-penrose.jpg?100}}]]                                                                               | **The Road to Reality: A Complete Guide to the Laws of the Universe (2004)**\\ //Roger Penrose//\\ Roger Penrose presents the only comprehensive and comprehensible account of the physics of the universe. From the very first attempts by the Greeks to grapple with the complexities of our known world to the latest application of infinity in physics, The Road to Reality carefully explores the movement of the smallest atomic particles and reaches into the vastness of intergalactic space. Here, Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | | [[https://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311|{{:ga:the_road_to_reality-penrose.jpg?100}}]]                                                                               | **The Road to Reality: A Complete Guide to the Laws of the Universe (2004)**\\ //Roger Penrose//\\ Roger Penrose presents the only comprehensive and comprehensible account of the physics of the universe. From the very first attempts by the Greeks to grapple with the complexities of our known world to the latest application of infinity in physics, The Road to Reality carefully explores the movement of the smallest atomic particles and reaches into the vastness of intergalactic space. Here, Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
-| [[https://www.amazon.com/How-Schrodingers-Cat-Escaped-Box/dp/9814635197|{{:ga:how_schrodinger_cat_escaped_the_box-rowlands.jpg?100}}]]                                                                  | **How Schrodinger's Cat Escaped the Box (2015)**\\ //Peter Rowlands//\\ This book attempts to explain the core of physics, the origin of everything and anything. It explains why physics at the most fundamental level, and especially quantum mechanics, has moved away from naive realism towards abstraction, and how this means that we can begin to answer some of the most fundamental questions which trouble us all, about space, time, matter, etc. It provides an original approach based on symmetry which will be of interest to professionals as well as lay people.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +| [[https://www.amazon.com/How-Schrodingers-Cat-Escaped-Box/dp/9814635197|{{:ga:how_schrodinger_cat_escaped_the_box-rowlands.jpg?100}}]] | **How Schrodinger's Cat Escaped the Box (2015)**\\ //Peter Rowlands//\\ This book attempts to explain the core of physics, the origin of everything and anything. It explains why physics at the most fundamental level, and especially quantum mechanics, has moved away from naive realism towards abstraction, and how this means that we can begin to answer some of the most fundamental questions which trouble us all, about space, time, matter, etc. It provides an original approach based on symmetry which will be of interest to professionals as well as lay people.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | 
-| [[https://www.amazon.com/Foundations-Game-Engine-Development-Mathematics/dp/0985811749|{{:ga:foundations_of_game_engine_development-lengyel.jpg?100}}]]                                                 | **Foundations of Game Engine Development, Volume 1: Mathematics (2016)**\\ //Eric Lengyel//\\ The goal of this project is to create a new textbook that provides a detailed introduction to the mathematics used by modern game engine programmers. (...) One example of an important topic missing from the existing literature is Grassmann algebra. Ever since I began giving conference presentations on this topic in 2012, I have been frequently asked whether there is a book that provides a thorough and intuitive discussion of Grassmann algebra (or the related topic of geometric algebra) in the context of game development and computer graphics. I don't believe that there is, and I'd like to take the opportunity to fill this void.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +| [[https://www.amazon.com/Foundations-Game-Engine-Development-Mathematics/dp/0985811749|{{:ga:foundations_of_game_engine_development-lengyel.jpg?100}}]] | **Foundations of Game Engine Development, Volume 1: Mathematics (2016)**\\ //Eric Lengyel//\\ The goal of this project is to create a new textbook that provides a detailed introduction to the mathematics used by modern game engine programmers. (...) One example of an important topic missing from the existing literature is Grassmann algebra. Ever since I began giving conference presentations on this topic in 2012, I have been frequently asked whether there is a book that provides a thorough and intuitive discussion of Grassmann algebra (or the related topic of geometric algebra) in the context of game development and computer graphics. I don't believe that there is, and I'd like to take the opportunity to fill this void.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | 
-| [[https://www.amazon.com/Invariant-Algebras-Geometric-Reasoning-Hongbo/dp/9812708081|{{:ga:invariant_algebras_and_geometric_reasoning-li.jpg?100}}]]                                                    | **Invariant Algebras And Geometric Reasoning (2008)**\\ //Hongbo Li//\\ The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics among them, Grassmann-Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries. This book contains the author and his collaborators' most recent, original development of Grassmann-Cayley algebra and Geometric Algebra and their applications in automated reasoning of classical geometries. It includes two of the three advanced invariant algebras Cayley bracket algebra, conformal geometric algebra, and null bracket algebra for highly efficient geometric computing. They form the theory of advanced invariants, and capture the intrinsic beauty of geometric languages and geometric computing.                                                                                                                                                                                                                                                                                                                                                                                                                                              | +| [[https://www.amazon.com/Foundations-Game-Engine-Development-Rendering/dp/0985811757|{{:ga:foundations_of_game_engine_development2-lengyel.jpg?100}}]] | **Foundations of Game Engine Development, Volume 2: Rendering (2019)**\\ //Eric Lengyel//\\ This second volume in the Foundations of Game Engine Development series explores the vast subject of real-time rendering in modern game engines. The book provides a detailed introduction to color science, world structure, projections, shaders, lighting, shadows, fog, and visibility methods. This is followed by extensive discussions of a variety of advanced rendering techniques that include volumetric effects, atmospheric shadowing, ambient occlusion, motion blur, and isosurface extraction.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | 
-| [[https://www.amazon.com/Code-Generation-Geometric-Algebra-Abstractions/dp/3330804653|{{:ga:code_generation_for_geometric_algebra-eid.jpg?100}}]]                                                       | **Code Generation for Geometric Algebra: Combining Geometric, Algebraic, and Software Abstractions for Scientific Computations (2016)**\\ //Ahmad Hosny Eid//\\ Geometry is an essential part of almost every solution to problems in computer science and engineering. The process of manually creating geometry-enabled software for representing and processing geometric data is tedious and error-prone. The use of automatic code generators for such process is more suitable, especially for low-dimensional geometric problems. Such problems can be commonly found in computer graphics, computer vision, robotics, and many other fields of application. A mathematical algebraic system with special characteristics must be used as the base of the code transformations required for final code generation. Geometric Algebra (GA) is one of the most suitable algebraic systems for such task. This work describes the design and use of one such GA-based prototype code generation system: GMac.                                                                                                                                                                                                                                                                                                                                                                   | +| [[https://www.amazon.com/Invariant-Algebras-Geometric-Reasoning-Hongbo/dp/9812708081|{{:ga:invariant_algebras_and_geometric_reasoning-li.jpg?100}}]] | **Invariant Algebras And Geometric Reasoning (2008)**\\ //Hongbo Li//\\ The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics among them, Grassmann-Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries. This book contains the author and his collaborators' most recent, original development of Grassmann-Cayley algebra and Geometric Algebra and their applications in automated reasoning of classical geometries. It includes two of the three advanced invariant algebras Cayley bracket algebra, conformal geometric algebra, and null bracket algebra for highly efficient geometric computing. They form the theory of advanced invariants, and capture the intrinsic beauty of geometric languages and geometric computing.                                                                                                                                                                                                                                                                                                                                                                                                                                              | 
-| [[http://www.xtec.cat/~rgonzal1/treatise2.htm|{{:ga:treatise_of_plane_geometry_through_geometric_algebra-gonzalez-calvet.jpg?100}}]]                                                                    | **Treatise of Plane Geometry through Geometric Algebra (2007)**\\ //Ramon González Calvet//\\ The Clifford-Grassman Geometric Algebra is applied to solve geometric equations, which are like the algebraic equations but containing geometric (vector) unknowns instead of real quantities. The unique way to solve these kind of equations is by using an associative algebra of vectors, the Clifford-Grassmann Geometric Algebra. Using the CGGA we have the freedom of transposition and isolation of any geometric unknown in a geometric equation. Then the typical problems and theorems of geometry, which have been hardly proved till now by means of synthetic methods, are more easily solved through the CGGA. On the other hand, the formulas obtained from the CGGA can immediately be written in Cartesian coordinates, giving a very useful service for programming computer applications, as in the case of the notable points of a triangle.                                                                                                                                                                                                                                                                                                                                                                                                                    | +| [[https://www.amazon.com/Code-Generation-Geometric-Algebra-Abstractions/dp/3330804653|{{:ga:code_generation_for_geometric_algebra-eid.jpg?100}}]] | **Code Generation for Geometric Algebra: Combining Geometric, Algebraic, and Software Abstractions for Scientific Computations (2016)**\\ //Ahmad Hosny Eid//\\ Geometry is an essential part of almost every solution to problems in computer science and engineering. The process of manually creating geometry-enabled software for representing and processing geometric data is tedious and error-prone. The use of automatic code generators for such process is more suitable, especially for low-dimensional geometric problems. Such problems can be commonly found in computer graphics, computer vision, robotics, and many other fields of application. A mathematical algebraic system with special characteristics must be used as the base of the code transformations required for final code generation. Geometric Algebra (GA) is one of the most suitable algebraic systems for such task. This work describes the design and use of one such GA-based prototype code generation system: GMac.                                                                                                                                                                                                                                                                                                                                                                   | 
-| [[https://www.amazon.com/Modern-Mathematics-Applications-Computer-Graphics/dp/9814449326|{{:ga:modern_mathematics_and_applications_in_computer_graphics_and_vision-guo.jpg?100}}]]                      | **Modern Mathematics and Applications in Computer Graphics and Vision (2014)**\\ //Hongyu Guo//\\ Presents a concise exposition of modern mathematical concepts, models and methods with applications in computer graphics, vision and machine learning. The compendium is organized in four parts: Algebra, Geometry, Topology, and Applications. One of the features is a unique treatment of tensor and manifold topics to make them easier for the students. All proofs are omitted to give an emphasis on the exposition of the concepts.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |+| [[http://www.xtec.cat/~rgonzal1/treatise2.htm|{{:ga:treatise_of_plane_geometry_through_geometric_algebra-gonzalez-calvet.jpg?100}}]] | **Treatise of Plane Geometry through Geometric Algebra (2007)**\\ //Ramon González Calvet//\\ The Clifford-Grassman Geometric Algebra is applied to solve geometric equations, which are like the algebraic equations but containing geometric (vector) unknowns instead of real quantities. The unique way to solve these kind of equations is by using an associative algebra of vectors, the Clifford-Grassmann Geometric Algebra. Using the CGGA we have the freedom of transposition and isolation of any geometric unknown in a geometric equation. Then the typical problems and theorems of geometry, which have been hardly proved till now by means of synthetic methods, are more easily solved through the CGGA. On the other hand, the formulas obtained from the CGGA can immediately be written in Cartesian coordinates, giving a very useful service for programming computer applications, as in the case of the notable points of a triangle.                                                                                                                                                                                                                                                                                                                                                                                                                    | 
 +| [[https://www.amazon.com/Modern-Mathematics-Applications-Computer-Graphics/dp/9814449326|{{:ga:modern_mathematics_and_applications_in_computer_graphics_and_vision-guo.jpg?100}}]] | **Modern Mathematics and Applications in Computer Graphics and Vision (2014)**\\ //Hongyu Guo//\\ Presents a concise exposition of modern mathematical concepts, models and methods with applications in computer graphics, vision and machine learning. The compendium is organized in four parts: Algebra, Geometry, Topology, and Applications. One of the features is a unique treatment of tensor and manifold topics to make them easier for the students. All proofs are omitted to give an emphasis on the exposition of the concepts.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
 | [[http://www.morikita.co.jp/books/book/2745|{{:ga:geometric_algebra-kanaya.jpg?100}}]]                                                                                                                  | **幾何学と代数系 Geometric Algebra (2014)**\\ //金谷 健一//\\ アメリカの物理学者ヘステネスを中心に提唱された「幾何学的代数」(geometric algebra) は,幾何学に古典的な代数系を対応させる手法であり,現在,物理学や工学のさまざまな分野で関心が寄せられている.本書は,この幾何学的代数の和書初となる入門書である.まず,背景をなすハミルトン代数,グラスマン代数,クリフォード代数を初歩からていねいに解説しているため,初学者でも自然に幾何学的代数の考え方を学ぶことができる.また,現代数学とのつながりも随所に見せることで,より深い理解が得られるようになっている.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | | [[http://www.morikita.co.jp/books/book/2745|{{:ga:geometric_algebra-kanaya.jpg?100}}]]                                                                                                                  | **幾何学と代数系 Geometric Algebra (2014)**\\ //金谷 健一//\\ アメリカの物理学者ヘステネスを中心に提唱された「幾何学的代数」(geometric algebra) は,幾何学に古典的な代数系を対応させる手法であり,現在,物理学や工学のさまざまな分野で関心が寄せられている.本書は,この幾何学的代数の和書初となる入門書である.まず,背景をなすハミルトン代数,グラスマン代数,クリフォード代数を初歩からていねいに解説しているため,初学者でも自然に幾何学的代数の考え方を学ぶことができる.また,現代数学とのつながりも随所に見せることで,より深い理解が得られるようになっている.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
-| [[https://www.amazon.com/Exterior-Algebras-Elementary-Tribute-Grassmanns/dp/1785482378|{{:ga:exterior_algebras-pavan.jpg?100}}]]                                                                        | **Exterior Algebras: Elementary Tribute to Grassmann's Ideas (2017)**\\ //Vincent Pavan//\\ Provides the theoretical basis for exterior computations. It first addresses the important question of constructing (pseudo)-Euclidian Grassmmann's algebras. Then, it shows how the latter can be used to treat a few basic, though significant, questions of linear algebra, such as co-linearity, determinant calculus, linear systems analyzing, volumes computations, invariant endomorphism considerations, skew-symmetric operator studies and decompositions, and Hodge conjugation, amongst others. Presents a self-contained guide that does not require any specific algebraic background. Includes numerous examples and direct applications that are suited for beginners.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +| [[https://www.amazon.com/Exterior-Algebras-Elementary-Tribute-Grassmanns/dp/1785482378|{{:ga:exterior_algebras-pavan.jpg?100}}]] | **Exterior Algebras: Elementary Tribute to Grassmann's Ideas (2017)**\\ //Vincent Pavan//\\ Provides the theoretical basis for exterior computations. It first addresses the important question of constructing (pseudo)-Euclidian Grassmmann's algebras. Then, it shows how the latter can be used to treat a few basic, though significant, questions of linear algebra, such as co-linearity, determinant calculus, linear systems analyzing, volumes computations, invariant endomorphism considerations, skew-symmetric operator studies and decompositions, and Hodge conjugation, amongst others. Presents a self-contained guide that does not require any specific algebraic background. Includes numerous examples and direct applications that are suited for beginners.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | 
-| [[https://www.amazon.com/dp/331990664X|{{:ga:a_geometric_algebra_invitation-lavor_xambo.jpg?100}}]]                                                                                                     | **A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry (2018)**\\ //Carlile Lavor, Sebastià Xambó-Descamps//\\ This book offers a gentle introduction to key elements of Geometric Algebra, along with their applications in Physics, Robotics and Molecular Geometry. Major applications covered are the physics of space-time, including Maxwell electromagnetism and the Dirac equation; robotics, including formulations for the forward and inverse kinematics and an overview of the singularity problem for serial robots; and molecular geometry, with 3D-protein structure calculations using NMR data. The book is primarily intended for graduate students and advanced undergraduates in related fields, but can also benefit professionals in search of a pedagogical presentation of these subjects.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +| [[https://www.amazon.com/dp/331990664X|{{:ga:a_geometric_algebra_invitation-lavor_xambo.jpg?100}}]] | **A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry (2018)**\\ //Carlile Lavor, Sebastià Xambó-Descamps//\\ This book offers a gentle introduction to key elements of Geometric Algebra, along with their applications in Physics, Robotics and Molecular Geometry. Major applications covered are the physics of space-time, including Maxwell electromagnetism and the Dirac equation; robotics, including formulations for the forward and inverse kinematics and an overview of the singularity problem for serial robots; and molecular geometry, with 3D-protein structure calculations using NMR data. The book is primarily intended for graduate students and advanced undergraduates in related fields, but can also benefit professionals in search of a pedagogical presentation of these subjects.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | 
-| [[https://www.amazon.com/dp/3319748289|{{:ga:geometric-algebra-applications_vol_i-bayro.jpg?100}}]]                                                                                                     | **Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing (2018)**\\ //Eduardo Bayro-Corrochano//\\ The goal is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +| [[https://www.amazon.com/dp/3319748289|{{:ga:geometric-algebra-applications_vol_i-bayro.jpg?100}}]] | **Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing (2018)**\\ //Eduardo Bayro-Corrochano//\\ The goal is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
-| [[https://www.amazon.com/dp/1498748384|{{:ga:introduction_to_geometric_algebra_computing-hildenbrand.jpg?100}}]]                                                                                        | **Introduction to Geometric Algebra Computing (Computer Vision) (2018)**\\ //Dietmar Hildenbrand//\\ This book is intended to give a rapid introduction to computing with Geometric Algebra and its power for geometric modeling. From the geometric objects point of view, it focuses on the most basic ones, namely points, lines and circles. This algebra is called Compass Ruler Algebra, since it is comparable to working with a compass and ruler. The book explores how to compute with these geometric objects, and their geometric operations and transformations, in a very intuitive way. The book follows a top-down approach, and while it focuses on 2D, it is also easily expandable to 3D computations. Algebra in engineering applications such as computer graphics, computer vision and robotics are also covered.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |+| [[https://www.amazon.com/dp/3030349764|{{:ga:geometric-algebra-applications_vol_ii-bayro.jpg?100}}]] | **Geometric Algebra Applications Vol. II: Robot Modelling and Control (2020)**\\ //Eduardo Bayro-Corrochano//\\ This book presents a unified mathematical treatment of diverse problems in the general domain of robotics and associated fields using Clifford or geometric algebra. By addressing a wide spectrum of problems in a common language, it offers both fresh insights and new solutions that are useful to scientists and engineers working in areas related with robotics. It introduces non-specialists to Clifford and geometric algebra, and provides examples to help readers learn how to compute using geometric entities and geometric formulations. It also includes an in-depth study of applications of Lie group theory, Lie algebra, spinors and versors and the algebra of incidence using the universal geometric algebra generated by reciprocal null cones. Featuring a detailed study of kinematics, differential kinematics and dynamics using geometric algebra, the book also develops Euler Lagrange and Hamiltonians equations for dynamics using conformal geometric algebra, and the recursive Newton-Euler using screw theory in the motor algebra framework. Further, it comprehensively explores robot modeling and nonlinear controllers, and discusses several applications in computer vision, graphics, neurocomputing, quantum computing, robotics and control engineering using the geometric algebra framework.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
 +| [[https://www.amazon.com/dp/1498748384|{{:ga:introduction_to_geometric_algebra_computing-hildenbrand.jpg?100}}]] | **Introduction to Geometric Algebra Computing (Computer Vision) (2018)**\\ //Dietmar Hildenbrand//\\ This book is intended to give a rapid introduction to computing with Geometric Algebra and its power for geometric modeling. From the geometric objects point of view, it focuses on the most basic ones, namely points, lines and circles. This algebra is called Compass Ruler Algebra, since it is comparable to working with a compass and ruler. The book explores how to compute with these geometric objects, and their geometric operations and transformations, in a very intuitive way. The book follows a top-down approach, and while it focuses on 2D, it is also easily expandable to 3D computations. Algebra in engineering applications such as computer graphics, computer vision and robotics are also covered.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/dp/3319946366|{{:ga:imaginary_mathematics_for_computer_science-vince.jpg?100}}]] | **Imaginary Mathematics for Computer Science (2018)**\\ //John Vince//\\ The imaginary unit //i// has been used by mathematicians for nearly five-hundred years, during which time its physical meaning has been a constant challenge. Unfortunately, Rene Descartes referred to it as "imaginary", and the use of the term "complex number" compounded the unnecessary mystery associated with this amazing object. Today, //i// has found its way into virtually every branch of mathematics, and is widely employed in physics and science, from solving problems in electrical engineering to quantum field theory. John Vince describes the evolution of the imaginary unit from the roots of quadratic and cubic equations, Hamilton's quaternions, Cayley's octonions, to Grassmann's geometric algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/dp/1498738915|{{:ga:handbook_of_geometric_constraint_systems_principles-crc.jpg?100}}]] | **Handbook of Geometric Constraint Systems Principles (2018)**\\ //Meera Sitharam, Audrey St. John, Jessica Sidman//\\ Entry point to the currently used principal mathematical and computational tools and techniques of the geometric constraint system (GCS). It functions as a single source containing the core principles and results, accessible to both beginners and experts. The handbook provides a guide for students learning basic concepts, as well as experts looking to pinpoint specific results or approaches in the broad landscape. As such, the editors created this handbook to serve as a useful tool for navigating the varied concepts, approaches and results found in GCS research.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/dp/0815378688|{{:ga:neural_networks_for_robotics-crc.jpg?100}}]] | **Neural Networks for Robotics: An Engineering Perspective (2018)**\\ //Nancy Arana-Daniel, Alma Y. Alanis, Carlos Lopez-Franco//\\ The book offers an insight on artificial neural networks for giving a robot a high level of autonomous tasks, such as navigation, cost mapping, object recognition, intelligent control of ground and aerial robots, and clustering, with real-time implementations. The reader will learn various methodologies that can be used to solve each stage on autonomous navigation for robots, from object recognition, clustering of obstacles, cost mapping of environments, path planning, and vision to low level control. These methodologies include real-life scenarios to implement a wide range of artificial neural network architectures.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/Introduction-Theoretical-Kinematics-mathematics-movement/dp/0978518039|{{:ga:introduction_to_theoretical_kinematics-mccarthy.jpg?100}}]] | **Introduction to Theoretical Kinematics: The mathematics of movement (2018)**\\ //J. Michael McCarthy//\\ An introduction to the mathematics used to model the articulated movement of mechanisms, machines, robots, and human and animal skeletons. Its concise and readable format emphasizes the similarity of the mathematics for planar, spherical and spatial movement. A modern approach introduces Lie groups and algebras and uses the theory of multivectors and Clifford algebras to clarify the construction of screws and dual quaternions. The pursuit of a unified presentation has led to a format in which planar, spherical, and spatial mechanisms are all studied in each chapter.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/Real-Spinorial-Groups-Mathematical-SpringerBriefs/dp/3030004031|{{:ga:real_spinorial_groups-xambo.jpg?100}}]] | **Real Spinorial Groups: A Short Mathematical Introduction (2018)**\\ //Sebastià Xambó-Descamps//\\ This book explores the Lipschitz spinorial groups (versor, pinor, spinor and rotor groups) of a real non-degenerate orthogonal geometry and how they relate to the group of isometries of that geometry. Once it has established the language of geometric algebra (linear grading of the algebra; geometric, exterior and interior products; involutions), it defines the spinorial groups, demonstrates their relation to the isometry groups, and illustrates their suppleness (geometric covariance) with a variety of examples.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/Topics-Clifford-Analysis-Wolfgang-Mathematics/dp/3030238539|{{:ga:topics_in_clifford_analysis-bernstein.jpg?100}}]] | **Topics in Clifford Analysis: Special Volume in Honor of Wolfgang Sprößig (2019)**\\ // Swanhild Bernstein (Editor)//\\ Quaternionic and Clifford analysis are an extension of complex analysis into higher dimensions. The unique starting point of Wolfgang Sprößig's work was the application of quaternionic analysis to elliptic differential equations and boundary value problems. The aim of this volume is to provide an essential overview of modern topics in Clifford analysis, presented by specialists in the field, and to honor the valued contributions to Clifford analysis made by Wolfgang Sprößig throughout his career.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/Geometric-Multivector-Analysis-Birkh%C3%A4user-Lehrb%C3%BCcher/dp/3030314103|{{:ga:geometric_multivector_analysis-rosen.jpg?100}}]] | **Geometric Multivector Analysis: From Grassmann to Dirac (2019)**\\ // Andreas Rosén//\\ Presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of Marcel Riesz and Lars Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions. Also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/Advanced-Calculus-Fundamentals-Carlos-Polanco/dp/9811415072|{{:ga:advanced_calculus-polanco.jpg?100}}]] | **Advanced Calculus - Fundamentals of Mathematics (2019)**\\ // Carlos Polanco//\\ Vector calculus is an essential mathematical tool for performing mathematical analysis of physical and natural phenomena. It is employed in advanced applications in the field of engineering and computer simulations.This textbook covers the fundamental requirements of vector calculus in curricula for college students in mathematics and engineering programs. Chapters start from the basics of vector algebra, real valued functions, different forms of integrals, geometric algebra and the various theorems relevant to vector calculus and differential forms.Readers will find a concise and clear study of vector calculus, along with several examples, exercises, and a case study in each chapter.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/Geometric-Multiplication-Vectors-Introduction-Mathematics/dp/3030017559|{{:ga:geometric_multiplication_of_vectors-josipovic.jpg?100}}]] | **Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics (2019)**\\ // Miroslav Josipović//\\ Enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra? What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics? What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science? In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics? Such a mathematical language exists and it is called geometric algebra. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/dp/1704596629|{{:ga:matrix-gateway-to-geometric-algebra_sobczyk.jpg?100}}]] | **Matrix Gateway to Geometric Algebra, Spacetime and Spinors (2019)**\\ // Garret Sobczyk//\\ Geometric algebra has been presented in many different guises since its invention by William Kingdon Clifford shortly before his death in 1879. In this book we fully integrate the ideas of geometric algebra directly into the fabric of matrix linear algebra. A geometric matrix is a real or complex matrix which is identified with a unique geometric number. The matrix product of two geometric matrices is just the product of the corresponding geometric numbers. Any equation can be either interpreted as a matrix equation or an equation in geometric algebra, thus fully unifying the two languages. The first 6 chapters provide an introduction to geometric algebra, and the classification of all such algebras. The last 3 chapters explore more advanced topics in the application of geometric algebras to Pauli and Dirac spinors, special relativity, Maxwell’s equations, quaternions, split quaternions, and group manifolds. They are included to highlight the great variety of topics that are imbued with new geometric insight when expressed in geometric algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
 +| [[https://www.amazon.com/Clifford-Algebras-Zeons-Geometry-Combinatorics/dp/9811202575|{{:ga:clifford_algebras_and_zeons-staples.jpg?100}}]] | **Clifford Algebras And Zeons: Geometry to Combinatorics and Beyond (2020)**\\ // George Stacey Staples//\\ Clifford algebras have many well-known applications in physics, engineering, and computer graphics. Zeon algebras are subalgebras of Clifford algebras whose combinatorial properties lend them to graph-theoretic applications such as enumerating minimal cost paths in dynamic networks. This book provides a foundational working knowledge of zeon algebras, their properties, and their potential applications in an increasingly technological world. As the first textbook to explore algebraic and combinatorial properties of zeon algebras in depth, it is suitable for interdisciplinary study in analysis, algebra, and combinatorics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | 
geometric_algebra.1528449922.txt.gz · Last modified: 2018/06/08 09:25 by pbk

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