geometric_algebra

This shows you the differences between two versions of the page.

Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||

geometric_algebra [2018/11/11 05:10] pbk [Articles] |
geometric_algebra [2019/09/25 23:12] pbk [Research] |
||
---|---|---|---|

Line 15: | Line 15: | ||

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. | ||

Line 24: | Line 24: | ||

* [[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 | * [[https://scholar.google.com/scholar?q="geometric+algebra"+OR+"clifford+algebra"|Geometric Algebra OR Clifford Algebra]] at Google Scholar | ||

Line 34: | Line 36: | ||

* [[https://www.quora.com/topic/Geometric-Algebra|Geometric Algebra]] topic at Quora | * [[https://www.quora.com/topic/Geometric-Algebra|Geometric Algebra]] topic at Quora | ||

+ | |||

+ | * [[https://observablehq.com/search?query=geometric%20algebra|Geometric Algebra]] at Observable | ||

* [[https://duckduckgo.com/c/Geometric_algebra|Geometric Algebra]] topic at DuckDuckGo | * [[https://duckduckgo.com/c/Geometric_algebra|Geometric Algebra]] topic at DuckDuckGo | ||

Line 40: | Line 44: | ||

* [[https://ncatlab.org/nlab/show/Ausdehnungslehre|Ausdehnungslehre]] | * [[https://ncatlab.org/nlab/show/Ausdehnungslehre|Ausdehnungslehre]] | ||

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. | ||

+ | |||

+ | * [[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 | * [[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// | ||

Line 60: | Line 71: | ||

* [[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// | ||

Line 79: | Line 91: | ||

> 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 scratch, instead of defining quaternions out of nowhere and trying to explain how they work retroactively. | > 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 scratch, instead of defining quaternions out of nowhere and trying to explain how they work retroactively. | ||

+ | |||

+ | * [[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 ==== | ||

Line 93: | Line 109: | ||

* [[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]] by //Charles Gunn//. | ||

==== Book companion ==== | ==== Book companion ==== | ||

Line 98: | Line 116: | ||

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. | ||

Line 122: | Line 140: | ||

* [[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/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]] | ||

Line 168: | Line 187: | ||

* [[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//. | ||

Line 189: | Line 209: | ||

* [[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//. | ||

Line 214: | Line 234: | ||

* [[http://www.gary-harper.com|Ripples in Space-Fabric]] - //Gary Harper//. | * [[http://www.gary-harper.com|Ripples in Space-Fabric]] - //Gary Harper//. | ||

* [[https://sites.google.com/site/samsilvaunesp|Samuel da Silva]] - Universidade Estadual Paulista. | * [[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. | ||

==== 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 | ||

Line 237: | Line 262: | ||

* [[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://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. | ||

===== Videos ===== | ===== Videos ===== | ||

Line 261: | Line 289: | ||

* [[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 270: | Line 298: | ||

* [[https://www.youtube.com/watch?v=ikCIUzX9myY|Joan Lasenby on Applications of Geometric Algebra in Engineering]] - //Y Combinator//. | * [[https://www.youtube.com/watch?v=ikCIUzX9myY|Joan Lasenby on Applications of Geometric Algebra in Engineering]] - //Y Combinator//. | ||

* [[https://www.youtube.com/watch?v=syyK6hTWT7U|Let's remove Quaternions from every 3D Engine]] - //Marc ten Bosch//. | * [[https://www.youtube.com/watch?v=syyK6hTWT7U|Let's remove Quaternions from every 3D Engine]] - //Marc ten Bosch//. | ||

+ | * [[https://www.youtube.com/watch?v=hbhxRM_YMv0|Overview of Geometric Algebra by Dr. Jack Hanlon]] - //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|A Brief introduction to Clifford Algebras by Johannes Familton]] - JMM2018 Quaternion Session, //Quaternion Notices//. | ||

+ | |||

===== Computing frameworks ===== | ===== Computing frameworks ===== | ||

Line 286: | Line 319: | ||

* [[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//. | ||

+ | * [[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://library.wolfram.com/infocenter/Conferences/6951|Symbolic and Numeric Geometric Algebra]] Mathematica notebook - //Terje Vold//. | ||

* [[http://nklein.com/software/geoma|Geoma]] - //Patrick Stein//. | * [[http://nklein.com/software/geoma|Geoma]] - //Patrick Stein//. | ||

Line 294: | Line 329: | ||

* [[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, //Lars Tingelstad//. Framework for estimation of multivectors in geometric algebra with focus on the Euclidean and conformal model. | ||

* [[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//. | ||

- | * [[http://gaonline.azurewebsites.net|GAonline]] - A flask, clifford and threejs/javascript visualiser for (4,1) Conformal Geometric Algebra (CGA), //Hugo Hadfield//. | + | * [[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/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/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/stephenathel/gawxm|GAwxM]] - Geometric Algebra using wxMaxima, //Stephen Abbott//. | ||

+ | * [[https://github.com/chakravala/Grassmann.jl|Grassmann.jl]] - Grassmann-Clifford-Hestenes-Taylor differential geometric algebra of hyper-dual multivector forms Julia package, //Dream Scatter//. | ||

+ | * [[https://github.com/pygae|pygae]] - Pythonic Geometric Algebra Enthusiasts at GitHub. | ||

===== 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// | ||

Line 319: | Line 356: | ||

* [[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// | ||

Line 705: | Line 745: | ||

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. | ||

Line 726: | Line 766: | ||

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. | 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. | ||

- | * [[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. | ||

Line 761: | Line 801: | ||

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 geometry. In 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. | 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 geometry. In 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. | ||

- | * [[http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pdf|Computer Graphics using Conformal Geometric Algebra]] (2006) - //Richard Wareham// | + | * [[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. | ||

Line 779: | Line 819: | ||

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). 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. | 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. | ||

Line 785: | Line 825: | ||

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. | 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. | ||

Line 830: | Line 870: | ||

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 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. | 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. | ||

Line 908: | Line 948: | ||

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. | ||

Line 926: | Line 966: | ||

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. | ||

Line 983: | Line 1023: | ||

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. | ||

Line 1041: | Line 1081: | ||

* [[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// | ||

Line 1063: | Line 1106: | ||

* [[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// | ||

Line 1078: | Line 1124: | ||

* [[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// | ||

Line 1139: | Line 1188: | ||

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 | 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 | ||

- | * [[http://www.kurtnalty.com/RegressiveVsAntiWedge.pdf|Regressive Versus Antiwedge Products]] (2018) - //Kurt Nalty// | + | * [[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. | The Hestenes (1986) regressive product differs from the Lengyel antiwedge product. | ||

Line 1163: | Line 1212: | ||

Plücker Coordinates Using Projective Representation]] (2018) - //Vaclav Skala, Michal Smolik// | 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. | 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/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. | ||

===== Books ===== | ===== Books ===== | ||

Line 1236: | Line 1333: | ||

| [[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/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/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/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/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/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. | | ||

Line 1243: | Line 1341: | ||

| [[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/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/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/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/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/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. | | ||

Line 1257: | Line 1356: | ||

| [[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/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/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. | | ||

+ |

geometric_algebra.txt · Last modified: 2020/03/22 16:07 by pbk