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geometric_algebra [2020/07/19 17:23] – [Articles] pbkgeometric_algebra [2023/12/30 00:23] (current) – [Videos] pbk
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   * [[https://duckduckgo.com/c/Geometric_algebra|Geometric Algebra]] topic at DuckDuckGo   * [[https://duckduckgo.com/c/Geometric_algebra|Geometric Algebra]] topic at DuckDuckGo
 +
 +  * [[https://github.com/topics/geometric-algebra|Geometric Algebra]] topic at GitHub
  
 ==== General ==== ==== General ====
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   * [[http://www.cgs-network.org/cgi20|ENGAGE 2020]] (2020) - Empowering Novel Geometric Algebra for Graphics & Engineering Workshop, CGI 2020, Geneva (Switzerland).   * [[http://www.cgs-network.org/cgi20|ENGAGE 2020]] (2020) - Empowering Novel Geometric Algebra for Graphics & Engineering Workshop, CGI 2020, Geneva (Switzerland).
 +
 +  * [[https://bivector.net/game2023.html|GAME2023 - Geometric Algebra Mini Event]] (2023) - DAE - Kortrijk, Belgium.
 ==== Book companion ==== ==== Book companion ====
  
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   * [[https://en.wikipedia.org/wiki/Vector_space|Vector space]]   * [[https://en.wikipedia.org/wiki/Vector_space|Vector space]]
   * [[https://en.wikipedia.org/wiki/Quaternion|Quaternion]]   * [[https://en.wikipedia.org/wiki/Quaternion|Quaternion]]
 +  * [[https://en.wikipedia.org/wiki/Biquaternion|Biquaternion]]
   * [[https://en.wikipedia.org/wiki/Octonion|Octonion]]   * [[https://en.wikipedia.org/wiki/Octonion|Octonion]]
   * [[https://en.wikipedia.org/wiki/Spinor|Spinor]]   * [[https://en.wikipedia.org/wiki/Spinor|Spinor]]
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   * [[http://isites.harvard.edu/fs/docs/icb.topic1048774.files/clif_mein.pdf|Clifford algebras and Lie groups]] {{ga:clif_mein.pdf|(local)}}- //Eckhard Meinrenken//, University of Toronto.   * [[http://isites.harvard.edu/fs/docs/icb.topic1048774.files/clif_mein.pdf|Clifford algebras and Lie groups]] {{ga:clif_mein.pdf|(local)}}- //Eckhard Meinrenken//, University of Toronto.
   * [[https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009|Applied Geometric Algebra]] (2009) - //László Tisza//, MIT OpenCourseWare.   * [[https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009|Applied Geometric Algebra]] (2009) - //László Tisza//, MIT OpenCourseWare.
 +  * [[https://www.cefns.nau.edu/~schulz/grassmann.pdf|Theory and application of Grassmann Algebra]] (2011) - //William C. Schulz//, Northern Arizona University.
   * [[https://lyseo.edu.ouka.fi/cms7/sites/default/files/geometric_algebra-tl.pdf|What is geometric algebra?]] (2016) - //Teuvo Laurinolli//, Oulun Lyseon.   * [[https://lyseo.edu.ouka.fi/cms7/sites/default/files/geometric_algebra-tl.pdf|What is geometric algebra?]] (2016) - //Teuvo Laurinolli//, Oulun Lyseon.
   * [[https://www.zatlovac.eu/lecturenotes/GAIntroLagape.pdf|Geometric Algebra and Calculus: Unified Language for Mathematics and Physics]] (2018) - //Vaclav Zatloukal//, Czech Technical University in Prague.   * [[https://www.zatlovac.eu/lecturenotes/GAIntroLagape.pdf|Geometric Algebra and Calculus: Unified Language for Mathematics and Physics]] (2018) - //Vaclav Zatloukal//, Czech Technical University in Prague.
   * [[https://dspace.library.uu.nl/bitstream/handle/1874/383367/IntroductionToGeometricAlgebraV2.pdf|Introduction to Geometric Algebra, a powerful tool for mathematics and physics]] (2019) - //Denis Lamaker//, Universiteit Utrecht.   * [[https://dspace.library.uu.nl/bitstream/handle/1874/383367/IntroductionToGeometricAlgebraV2.pdf|Introduction to Geometric Algebra, a powerful tool for mathematics and physics]] (2019) - //Denis Lamaker//, Universiteit Utrecht.
 +  * [[https://mattferraro.dev/posts/geometric-algebra|What is the Inverse of a Vector?]] (2021) - //Matt Ferraro//.
  
 ===== Videos ===== ===== Videos =====
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   * [[https://skillsmatter.com/skillscasts/13986-geometric-algebra-in-fsharp|Geometric Algebra in F#]] - //Andrew Willshire//.   * [[https://skillsmatter.com/skillscasts/13986-geometric-algebra-in-fsharp|Geometric Algebra in F#]] - //Andrew Willshire//.
   * [[https://www.youtube.com/watch?v=tX4H_ctggYo|SIGGRAPH 2019: Geometric Algebra for Computer Graphics]] - //Charles Gunn// and //Steven De Keninck//.   * [[https://www.youtube.com/watch?v=tX4H_ctggYo|SIGGRAPH 2019: Geometric Algebra for Computer Graphics]] - //Charles Gunn// and //Steven De Keninck//.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjzcQ4eCVAntETNNVD2d5S79|GAME2020 - Geometric Algebra Mini Event]] - //DAE Kortrijk, Belgium//.
 +  * [[https://www.youtube.com/watch?v=60z_hpEAtD8|A Swift Introduction to Geometric Algebra]] and [[https://www.youtube.com/playlist?list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy|From Zero to Geo]] - //sudgylacmoe//.
 +  * [[https://www.youtube.com/watch?v=cKfC2ZBJulg|Projective Geometric Algebra for Paraxial Geometric Optics]] - // Katelyn Spadavecchia//.
 +  * [[https://www.youtube.com/watch?v=11sH9X0OO9Y&list=PLnpuwbuviU2j7OSnZdstP5_g1ejA32bYA|Geometric Algebra Lectures ]] - //Miroslav Josipović//.
 +  * [[https://www.youtube.com/watch?v=HGcBu4TQgRE|Quaternions and Clifford Algebra]] - //Q. J. Ge and Anurag Purwar//, Stony Brook University.
 +  * [[https://www.youtube.com/watch?v=LestlcDk6Iw|Foundations of Geometric Algebra Computing]] - Lecture at ICU Tokyo, //Dietmar Hildenbrand//.
 +  * [[https://www.youtube.com/watch?v=e5D7Bma9Vhw&list=PLxo3PbygE0PLdFFy_2b02JAaUsleFW8py|Geometric Algebra]] - First Course in STEMCstudio, //David Geo Holmes//.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjyWv5wLVV7QfeS_d8pwCPv_|AGACSE2021]] - Selected talks.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc|Plane-based Geometric Algebra Tutorial]] - Presentation at SIBGRAPI 2021, //Steven De Keninck and Leo Dorst//.
 +  * [[https://www.youtube.com/watch?v=hTa_gErsrtM|Geometric Algebra]] - Talk at SIGGRAPH 2022, //Alyn Rockwood and Dietmar Hildenbrand//.
 +  * [[https://www.youtube.com/watch?v=8n6GsKWznfY|Plane Based Geometric Algebra]] - Advanced Computational Applications of GA, //Leo Dorst and Steven De Keninck//.
 +  * [[https://www.youtube.com/watch?v=PGZNYGwsXTw|Why Geometric Algebra Should be in the Standard Linear Algebra Curriculum]] and [[https://www.youtube.com/watch?v=ISKJPmuZkbY|Fun Applications of Geometric Algebra]] - Presentations at [[https://pgadey.ca/seminar/|Parker Glynn-Adey seminars]], //Logan Lim//.
 +  * [[https://www.youtube.com/watch?v=VXziLgMIWf8|Geometric Clifford Algebra Networks and Clifford Neural Layers for PDE Modeling]] - Valence Labs, //Johannes Brandstetter//.
 +  * [[https://www.youtube.com/watch?v=nktgFWLy32U|Spinors for Beginners 11: What is a Clifford Algebra?]] - //eigenchris//.
 +  * [[https://www.youtube.com/watch?v=htYh-Tq7ZBI|Why can't you multiply vectors?]] - Talk at Dutch Game Day 2023, //Freya Holmér//.
 +  * [[https://www.youtube.com/watch?v=1AmeD0Vc8ow|GAME2023 Geometric Algebra Mini Event]] - Livestream.
 +  * [[https://www.youtube.com/watch?v=zgi-13F2Kec|Geometric Algebra: The Prequel]] - at GAME2023, //Steven De Keninck//.
 +  * [[https://www.youtube.com/watch?v=nPIRL-c88_E|Geometric Algebra Transformers: Revolutionizing Geometric Data]] - at Intelligent Systems Conference 2023, //Taco Cohen//.
  
 ===== Computing frameworks ===== ===== Computing frameworks =====
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   * [[http://www.siue.edu/~sstaple/index_files/research.html|CliffMath]] - Clifford algebra computations, including zeon, sym-Clifford, and idem-Clifford subalgebras, //George Stacey Staples//.   * [[http://www.siue.edu/~sstaple/index_files/research.html|CliffMath]] - Clifford algebra computations, including zeon, sym-Clifford, and idem-Clifford subalgebras, //George Stacey Staples//.
   * [[https://github.com/Prograf-UFF/TbGAL|TbGAL]] - Tensor-Based Geometric Algebra C++/Python Library, //Eduardo Vera Sousa, Leandro A. F. Fernandes//.   * [[https://github.com/Prograf-UFF/TbGAL|TbGAL]] - Tensor-Based Geometric Algebra C++/Python Library, //Eduardo Vera Sousa, Leandro A. F. Fernandes//.
 +  * [[https://github.com/markisus/g3|G3]] - A library for the Geometric Algebra of the Vector Space R^3, //Markisus//.
 +  * [[https://github.com/vincentnozick/garamon|Garamon Generator]] - Geometric Algebra Recursive and Adaptative Monster is a generator of C++ libraries dedicated to Geometric Algebra, //Vincent Nozick, Stephane Breuils//.
 ===== Articles ===== ===== Articles =====
   * [[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf|Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics]] (2002) - //David Hestenes//   * [[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf|Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics]] (2002) - //David Hestenes//
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   * [[https://www.researchgate.net/publication/228955605_A_brief_introduction_to_Clifford_algebra|A brief introduction to Clifford Algebra]] (2010) - //Silvia Franchini, Giorgio Vassallo, Filippo Sorbello//   * [[https://www.researchgate.net/publication/228955605_A_brief_introduction_to_Clifford_algebra|A brief introduction to Clifford Algebra]] (2010) - //Silvia Franchini, Giorgio Vassallo, Filippo Sorbello//
 Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators. Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators.
 +
 +  * [[https://vixra.org/pdf/1203.0011v1.pdf|A Very Brief Introduction to Clifford Algebra]] (2012) - //Stephen Crowley//
 +This article distills many of the essential definitions from the very thorough book, Clifford Algebras: An Introduction, by Dr D.J.H. Garling, with some minor additions.
  
   * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf|An Introduction to Geometric Algebra and Calculus]] (2014) - //Alan Bromborsky//   * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf|An Introduction to Geometric Algebra and Calculus]] (2014) - //Alan Bromborsky//
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 This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper. This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper.
  
-  * [[https://sites.math.washington.edu/~morrow/336_17/papers17/josh.pdf|Article Review:  A Survey of Geometric Calculus and Geometric Algebra]] (2017) - //Josh Pollock//+  * [[https://sites.math.washington.edu/~morrow/336_17/papers17/josh.pdf|Article Review: A Survey of Geometric Calculus and Geometric Algebra]] (2017) - //Josh Pollock//
 In his article //A Survey of Geometric Calculus and Geometric Algebra//, Professor Alan Macdonald provides a brief introduction to geometric algebra (GA) and geometric calculus (GC) along with some applications to physics and a brief mention of the related projective and conformal geometric algebras. He only expects the reader to have knowledge of linear algebra and vector calculus. In this review, I hope to whet your appetite for GA and GC by showing some of its important results. In his article //A Survey of Geometric Calculus and Geometric Algebra//, Professor Alan Macdonald provides a brief introduction to geometric algebra (GA) and geometric calculus (GC) along with some applications to physics and a brief mention of the related projective and conformal geometric algebras. He only expects the reader to have knowledge of linear algebra and vector calculus. In this review, I hope to whet your appetite for GA and GC by showing some of its important results.
  
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   * [[https://www.informatik.uni-kiel.de/inf/Sommer/doc/Downloads/Publikationen/GeomComp.pdf|Geometric Computing with Clifford Algebras]] Theoretical Foundations and Applications in Computer Vision and Robotics (2001) - //Gerald Sommer//   * [[https://www.informatik.uni-kiel.de/inf/Sommer/doc/Downloads/Publikationen/GeomComp.pdf|Geometric Computing with Clifford Algebras]] Theoretical Foundations and Applications in Computer Vision and Robotics (2001) - //Gerald Sommer//
-This book presents a collection of contributions concerning the task of solving geometry related problems with suitable algebraic embeddings. It is not only directed at scientists who already discovered the power of Clifford algebras for their field, but also at those scientists who are interested in Clifford algebras and want to see how these can be applied to problems in computer science, signal theory, neural computation, computer vision and robotics. It was therefore tried to keep this book accessible to newcomers to applications of Clifford  algebra while still presenting up to date research and new developments.+This book presents a collection of contributions concerning the task of solving geometry related problems with suitable algebraic embeddings. It is not only directed at scientists who already discovered the power of Clifford algebras for their field, but also at those scientists who are interested in Clifford algebras and want to see how these can be applied to problems in computer science, signal theory, neural computation, computer vision and robotics. It was therefore tried to keep this book accessible to newcomers to applications of Clifford algebra while still presenting up to date research and new developments.
  
   * [[http://www.gaalop.de/dhilden_data/CLUScripts/eg04_tut03.pdf|Geometric Algebra and its Application to Computer Graphics]] (2004) - //D. Hildenbrand, D. Fontijne, C. Perwass, L. Dorst//   * [[http://www.gaalop.de/dhilden_data/CLUScripts/eg04_tut03.pdf|Geometric Algebra and its Application to Computer Graphics]] (2004) - //D. Hildenbrand, D. Fontijne, C. Perwass, L. Dorst//
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 A tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics. A tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics.
  
-  * [[https://pure.uva.nl/ws/files/4375498/52687_fontijne.pdf|Efficient Implementation of Geometric Algebra]]  (2007) - //Daniel Fontijne//+  * [[https://pure.uva.nl/ws/files/4375498/52687_fontijne.pdf|Efficient Implementation of Geometric Algebra]] (2007) - //Daniel Fontijne//
 This thesis addresses the computational and implementational aspects of geometric algebra, and shows that its mathematical promise can be made into programming reality: geometric algebra provides a modular, structured specification language for geometry whose implementations can be automatically generated, leading to an efficiency that is competitive with the (hand-) optimized code based on the traditional linear algebra approach. This thesis addresses the computational and implementational aspects of geometric algebra, and shows that its mathematical promise can be made into programming reality: geometric algebra provides a modular, structured specification language for geometry whose implementations can be automatically generated, leading to an efficiency that is competitive with the (hand-) optimized code based on the traditional linear algebra approach.
  
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   * [[http://www.scielo.org.mx/pdf/cys/v19n3/v19n3a6.pdf|Modeling and Pose Control of Robotic Manipulators and Legs using Conformal Geometric Algebra]] (2015) - //Oscar Carbajal-Espinosa et al//   * [[http://www.scielo.org.mx/pdf/cys/v19n3/v19n3a6.pdf|Modeling and Pose Control of Robotic Manipulators and Legs using Conformal Geometric Algebra]] (2015) - //Oscar Carbajal-Espinosa et al//
-Controlling the pose of a manipulator involves finding the correct configuration of the robot’s elements to move the end effector to a desired position and orientation. In  order  to  find  the  geometric  relationships between the elements of a robot manipulator, it is necessary to define the kinematics of the robot. We present a  synthesis  of the kinematical model of the pose for this type of robot using the conformal geometric algebra framework. In addition,  two controllers are developed, one for the position tracking problem and another for the orientation tracking problem, both using an error feedback controller. The stability analysis is carried out for both controllers, and their application to a 6-DOF serial manipulator and the legs of a biped robot are presented. By proposing the error feedback and Lyapunov functions in  terms  of  geometric  algebra,  we  are  opening   new venue of research in control of manipulators and robot legs that involves the use of geometric primitives, such as lines, circles, planes, spheres.+Controlling the pose of a manipulator involves finding the correct configuration of the robot’s elements to move the end effector to a desired position and orientation. In order to find the geometric relationships between the elements of a robot manipulator, it is necessary to define the kinematics of the robot. We present a synthesis of the kinematical model of the pose for this type of robot using the conformal geometric algebra framework. In addition, two controllers are developed, one for the position tracking problem and another for the orientation tracking problem, both using an error feedback controller. The stability analysis is carried out for both controllers, and their application to a 6-DOF serial manipulator and the legs of a biped robot are presented. By proposing the error feedback and Lyapunov functions in terms of geometric algebra, we are opening a new venue of research in control of manipulators and robot legs that involves the use of geometric primitives, such as lines, circles, planes, spheres.
  
   * [[https://www.researchgate.net/profile/Leo_Dorst/publication/254901215_Competitive_runtime_performance_for_inverse_kinematics_algorithms_using_conformal_geometric_algebra/links/5444bfb20cf2a76a3ccd81cd.pdf|Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra]] (2006) - //Dietmar Hildenbrand, Daniel Fontijne, Yusheng Wang, Marc Alexa, Leo Dorst//   * [[https://www.researchgate.net/profile/Leo_Dorst/publication/254901215_Competitive_runtime_performance_for_inverse_kinematics_algorithms_using_conformal_geometric_algebra/links/5444bfb20cf2a76a3ccd81cd.pdf|Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra]] (2006) - //Dietmar Hildenbrand, Daniel Fontijne, Yusheng Wang, Marc Alexa, Leo Dorst//
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   * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/GA-SIG/Papers/Report.pdf|A Covariant Approach to Geometry using Geometric Algebra]] (2004) - //Anthony Lasenby, Joan Lasenby, Richard Wareham//   * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/GA-SIG/Papers/Report.pdf|A Covariant Approach to Geometry using Geometric Algebra]] (2004) - //Anthony Lasenby, Joan Lasenby, Richard Wareham//
-This report aims to show that using the mathematical framework of conformal geometric algebra – a 5-dimensional representation of 3-dimensional space – we are able to provide an elegant covariant approach to 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.
  
   * [[https://rjw57.github.io/phd-thesis/rjw-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.
  
-  * [[http://home.deib.polimi.it/tubaro/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro//+  * [[https://tubaro.faculty.polimi.it/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro//
 In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible. In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible.
  
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   * [[https://arxiv.org/pdf/1601.06044.pdf|Geometric-Algebra LMS Adaptive Filter and its Application to Rotation Estimation]] (2016) - //Wilder B. Lopes, Anas Al-Nuaimi, Cassio G. Lopes//   * [[https://arxiv.org/pdf/1601.06044.pdf|Geometric-Algebra LMS Adaptive Filter and its Application to Rotation Estimation]] (2016) - //Wilder B. Lopes, Anas Al-Nuaimi, Cassio G. Lopes//
-This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel  GA  least-mean-squares  (GA-LMS)  adaptive  filter,  which inherits properties from standard adaptive filters and from GA, is  developed  to  recursively  estimate   rotor  (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows.  Moreover,  the  employed  GA/GC  framework  allows  for easily applying the resulting filter to estimating rotors in higher dimensions.+This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.
  
   * [[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.//   * [[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.//
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   * [[http://www.naturalspublishing.com/files/published/7c772c2o9nshz1.pdf|Ortogonal Approach for Haptic Rendering Algorithm based in Conformal Geometric Algebra]] (2014) - //Gabriel Sepulveda-Cervantes, Edgar A. Portilla-Flores//   * [[http://www.naturalspublishing.com/files/published/7c772c2o9nshz1.pdf|Ortogonal Approach for Haptic Rendering Algorithm based in Conformal Geometric Algebra]] (2014) - //Gabriel Sepulveda-Cervantes, Edgar A. Portilla-Flores//
-This work presents a novel method for haptic rendering contact force and surface properties for virtual objects using the Conformal Geometric Algebra orthogonal  decomposition approach. The mathematical representation of geometric primitives along with collision algorithms based on its mathematical properties is presented. The orthogonal decomposition of contact and interaction forces is achieved using the same framework and dynamic properties in both subspaces are rendered simultaneously. Comparing with vector calculus,  the Conformal  Geometric Algebra (CGA) approach  provides an easier and more intuitive way to deal with haptic rendering problems due to its inner properties and a simpler representation of geometric objects and linear transformation. The results of the evaluation of the method using a 3 DOF haptic device are presented.+This work presents a novel method for haptic rendering contact force and surface properties for virtual objects using the Conformal Geometric Algebra orthogonal decomposition approach. The mathematical representation of geometric primitives along with collision algorithms based on its mathematical properties is presented. The orthogonal decomposition of contact and interaction forces is achieved using the same framework and dynamic properties in both subspaces are rendered simultaneously. Comparing with vector calculus, the Conformal Geometric Algebra (CGA) approach provides an easier and more intuitive way to deal with haptic rendering problems due to its inner properties and a simpler representation of geometric objects and linear transformation. The results of the evaluation of the method using a 3 DOF haptic device are presented.
  
   * [[https://arxiv.org/pdf/0904.3349v1.pdf|An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota]] (2009) - //Henry Crapo//   * [[https://arxiv.org/pdf/0904.3349v1.pdf|An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota]] (2009) - //Henry Crapo//
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   * [[https://arxiv.org/pdf/1611.09182.pdf|Standard model physics from an algebra?]] (2015) - //Cohl Furey//   * [[https://arxiv.org/pdf/1611.09182.pdf|Standard model physics from an algebra?]] (2015) - //Cohl Furey//
-This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra R⊗C⊗H⊗O, the tensor product of the only four normed division algebras over the real numbers. Using nothing more than R⊗C⊗H⊗O acting  on  itself, we set out to find standard model particle representations: a task which occupies the remainder of this text.+This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself. We then focus on a special case by considering the algebra R⊗C⊗H⊗O, the tensor product of the only four normed division algebras over the real numbers. Using nothing more than R⊗C⊗H⊗O acting on itself, we set out to find standard model particle representations: a task which occupies the remainder of this text.
  
   * [[http://www.mdpi.com/2073-8994/8/9/92/pdf|Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra]] (2016) - //Juan C. Bravo, Manuel V. Castilla//   * [[http://www.mdpi.com/2073-8994/8/9/92/pdf|Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra]] (2016) - //Juan C. Bravo, Manuel V. Castilla//
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   * [[https://www.researchgate.net/publication/264423339_An_invitation_to_Clifford_Analysis|Una Invitación al Análisis de Clifford]] (2003) - //Richard Delanghe, Juan Bory-Reyes//   * [[https://www.researchgate.net/publication/264423339_An_invitation_to_Clifford_Analysis|Una Invitación al Análisis de Clifford]] (2003) - //Richard Delanghe, Juan Bory-Reyes//
-Una panorámica  de los tópicos principales y herramientas básicas del Análisis de  Clifford se presenta en este artículo, al mismo tiempo, las principales fórmulas integrales relacionadas con la integral tipo Cauchy --- y  su versión  singular --- son  analizadas en un contexto multidimensional, con el uso de las  técnicas de álgebras de  Clifford. Se incluyen también algunas notas históricas sobre el desarrollo de este campo de investigación.+Una panorámica de los tópicos principales y herramientas básicas del Análisis de Clifford se presenta en este artículo, al mismo tiempo, las principales fórmulas integrales relacionadas con la integral tipo Cauchy --- y su versión singular --- son analizadas en un contexto multidimensional, con el uso de las técnicas de álgebras de Clifford. Se incluyen también algunas notas históricas sobre el desarrollo de este campo de investigación.
  
   * [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano//   * [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano//
Line 999: Line 1029:
  
   * [[http://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IFAC_ICINCO_2007/ICINCO%202007/Area%202%20-%20Robotics%20and%20Automation/Volume%202/Short%20Papers/C2_496_Bayro-Corrochano.pdf|Geometric Advanced Techniques for Robot Grasping using Stereoscopic Vision]] (2007) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//   * [[http://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IFAC_ICINCO_2007/ICINCO%202007/Area%202%20-%20Robotics%20and%20Automation/Volume%202/Short%20Papers/C2_496_Bayro-Corrochano.pdf|Geometric Advanced Techniques for Robot Grasping using Stereoscopic Vision]] (2007) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//
-In this paper the authors propose geometric techniques to deal with the problem of grasping objects relying on their mathematical models.  For that we use the geometric algebra framework to formulate the kinematics of a three finger robotic hand. Our main objective is by formulating a kinematic control law to close the loop between perception and actions. This allows us to perform a smooth visually guided object grasping action.+In this paper the authors propose geometric techniques to deal with the problem of grasping objects relying on their mathematical models. For that we use the geometric algebra framework to formulate the kinematics of a three finger robotic hand. Our main objective is by formulating a kinematic control law to close the loop between perception and actions. This allows us to perform a smooth visually guided object grasping action.
  
   * [[http://downloads.hindawi.com/journals/abb/2011/728132.pdf|Robot Object Manipulation Using Stereoscopic Vision and Conformal Geometric Algebra]] (2011) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//   * [[http://downloads.hindawi.com/journals/abb/2011/728132.pdf|Robot Object Manipulation Using Stereoscopic Vision and Conformal Geometric Algebra]] (2011) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano//
Line 1036: Line 1066:
   * [[https://www.mit.edu/~fengt/282C.pdf|The Atiyah–Singer index theorem]] (2015) - //Dan Berwick-Evans, via Tony Feng//   * [[https://www.mit.edu/~fengt/282C.pdf|The Atiyah–Singer index theorem]] (2015) - //Dan Berwick-Evans, via Tony Feng//
 Lecture notes about the Atiyah-Singer index theorem. Lecture notes about the Atiyah-Singer index theorem.
 +
 +  * [[http://www.cs.ox.ac.uk/people/david.reutter/AtiyahSinger_Essay.pdf|The Heat Equation and the Atiyah-Singer Index Theorem]] (2015) - //David Reutter//
 +The Atiyah-Singer index theorem is a milestone of twentieth century mathematics. Roughly speaking, it relates a global analytical datum of a manifold --- the number of solutions of a certain linear PDE --- to an integral of local topological expressions over this manifold. The index theorem provided a link between analysis, geometry and topology, paving the way for many further applications along these lines.
  
   * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie//   * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie//
Line 1107: Line 1140:
  
   * [[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//   * [[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.+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 1307: Line 1340:
   * [[https://arxiv.org/pdf/1911.08658|Spinors of real type as polyforms and the generalized Killing equation]] (2019) - //Vicente Cortes, Calin Lazaroiu, C. S. Shahbazi//   * [[https://arxiv.org/pdf/1911.08658|Spinors of real type as polyforms and the generalized Killing equation]] (2019) - //Vicente Cortes, Calin Lazaroiu, C. S. Shahbazi//
 We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS_4 space-time. We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS_4 space-time.
 +
 +  * [[https://link.springer.com/content/pdf/10.1007/s00006-019-0987-7.pdf|Garamon: A Geometric Algebra Library Generator]] (2019) - //Stephane Breuils, Vincent Nozick, Laurent Fuchs//
 +This paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++/Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. 
  
   * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler//   * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler//
Line 1337: Line 1373:
   * [[https://arxiv.org/pdf/2007.04464.pdf|Deform, Cut and Tear a skinned model using Conformal Geometric Algebra]] (2020) - //Manos Kamarianakis, George Papagiannakis//   * [[https://arxiv.org/pdf/2007.04464.pdf|Deform, Cut and Tear a skinned model using Conformal Geometric Algebra]] (2020) - //Manos Kamarianakis, George Papagiannakis//
 We present a novel, integrated rigged character simulation framework in Conformal Geometric Algebra (CGA) that supports, for the first time, real-time cuts and tears, before and/or after the animation, while maintaining deformation topology. The purpose of using CGA is to lift several restrictions posed by current state-of-the-art character animation & deformation methods. Previous implementations originally required weighted matrices to perform deformations, whereas, in the current state-of-the-art, dual-quaternions handle both rotations and translations, but cannot handle dilations. CGA is a suitable extension of dual-quaternion algebra that amends these two major previous shortcomings: the need to constantly transmute between matrices and dual-quaternions as well as the inability to properly dilate a model during animation. Our CGA algorithm also provides easy interpolation and application of all deformations in each intermediate steps, all within the same geometric framework. Furthermore we also present two novel algorithms that enable cutting and tearing of the input rigged, animated model, while the output model can be further re-deformed. We present a novel, integrated rigged character simulation framework in Conformal Geometric Algebra (CGA) that supports, for the first time, real-time cuts and tears, before and/or after the animation, while maintaining deformation topology. The purpose of using CGA is to lift several restrictions posed by current state-of-the-art character animation & deformation methods. Previous implementations originally required weighted matrices to perform deformations, whereas, in the current state-of-the-art, dual-quaternions handle both rotations and translations, but cannot handle dilations. CGA is a suitable extension of dual-quaternion algebra that amends these two major previous shortcomings: the need to constantly transmute between matrices and dual-quaternions as well as the inability to properly dilate a model during animation. Our CGA algorithm also provides easy interpolation and application of all deformations in each intermediate steps, all within the same geometric framework. Furthermore we also present two novel algorithms that enable cutting and tearing of the input rigged, animated model, while the output model can be further re-deformed.
 +
 +  * [[https://www.researchgate.net/profile/Carlos-Muro/publication/339249543_Newton-Euler_Modeling_and_Control_of_a_Multi-copter_Using_Motor_Algebra_mathbfG_301G301/links/603d925f299bf1e0784d02bd/Newton-Euler-Modeling-and-Control-of-a-Multi-copter-Using-Motor-Algebra-mathbfG-3-0-1G3-0-1.pdf|Newton-Euler Modelling and Control of a Multicopter using Motor Algebra G^+_3,0,1]] (2020) - //Carlos A. Arellano-Muro, Guillermo Osuna-Gonzalez, et al//
 +In this work the dynamic model and the nonlinear control for a multi-copter have been developed using the geometric algebra framework specifically using the motor algebra G^+_3,0,1. The kinematics for the aircraft model and the dynamics based on Newton-Euler formalism are presented. Block-control technique is applied to the multi-copter model which involves super twisting control and an estimator of the internal dynamics for maneuvers away from the origin. The stability of the presented control scheme is proved. The experimental analysis shows that our non-linear controller law is able to reject external disturbances and to deal with parametric variations.
 +
 +  * [[http://i-us.ru/index.php/ius/article/download/13579/14098|Human action recognition method based on conformal geometric algebra and recurrent neural network]] (2020) - //Nguyen Nang Hung Van, Pham Minh Tuan et al//
 +The use of Conformal Geometric Algebra in order to extract features and simultaneously reduce the dimensionality of a dataset for human activity recognition using Recurrent Neural Network.
 +
 +  * [[https://arxiv.org/pdf/2107.00343.pdf|Explicit Baker-Campbell-Hausdorff-Dynkin formula for Spacetime via Geometric Algebra]] (2021) - //Joseph Wilson, Matt Visser//
 +We present a compact Baker-Campbell-Hausdorff-Dynkin formula for the composition of Lorentz transformations e^σi in the spin representation (a.k.a. Lorentz rotors) in terms of their generators σi. This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension ≤4, naturally generalising Rodrigues' formula for rotations in R3. In particular, it applies to Lorentz rotors within the framework of Hestenes' spacetime algebra, and provides an efficient method for composing Lorentz generators.
 +
 +  * [[https://arxiv.org/pdf/2107.03771.pdf|Graded Symmetry Groups: Plane and Simple]] (2021) - //Martin Roelfs, Steven De Keninck//
 +The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a graded structure on Lie groups, which is not evident in their matrix representation. By embracing this graded structure, the invariant decomposition theorem was proven: any composition of k linearly independent reflections can be decomposed into ⌈k/2⌉ commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi-Chasles' theorem as its 3D Euclidean special case.
 +
 +  * [[https://www.pacm.princeton.edu/sites/default/files/pacm_arjunmani_0.pdf|Representing Words in a Geometric Algebra]] (2021) - //Arjun Mani//
 +In this paper we introduce and motivate geometric algebra as a better representation for word embeddings. Next we describe how to implement the geometric product and interestingly show that neural networks can learn this product. We then introduce a model that represents words as objects in this algebra and benchmark it on large corpuses; our results show some promise on traditional word embedding tasks. Thus, we lay the groundwork for further investigation of geometric algebra in word embeddings.
 +
 +  * [[https://ietresearch.onlinelibrary.wiley.com/doi/pdfdirect/10.1049/cmu2.12188|An approach to adaptive filtering with variable step size based on geometric algebra]] (2021) - //Haiquan Wang, Yinmei He et al//
 +Recently, adaptive filtering algorithms have attracted much more attention in the field of signal processing. By studying the shortcoming of the traditional real-valued fixed step size adaptive filtering algorithm, this paper proposed the novel approach to adaptive filtering with variable step size based on Sigmoid function and geometric algebra (GA).
 +
 +  * [[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9488174|A Survey on Quaternion Algebra and Geometric Algebra Applications in Engineering and Computer Science 1995–2020]] (2021) - //Eduardo Bayro-Corrochano//
 +Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020.
 +
 +  * [[https://dspace.library.uu.nl/bitstream/handle/1874/403340/thesis.pdf|Clifford algebras and their application in the Dirac equation]] (2021) - //Paul van Hoegaerden//
 +The aim of this thesis will be to study the Clifford algebras that appear in the derivation of the Dirac equation and investigate alternative formulations of the Dirac equation using (complex) quaternions. To this end, we will first look at the symmetries of the Dirac equation and some of the additional insights that follow from the Dirac equation. We will also give a derivation of the Dirac equation starting from the Schrödinger equation, in which we will come across the gamma matrices.
 +
 +  * [[https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/cta.3132|Geometric Algebra for teaching AC Circuit Theory]] (2021) - //Francisco G. Montoya, Raúl Baños et al//
 +This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non-sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses.
 ===== Books ===== ===== Books =====
  
 ==== Historical ==== ==== Historical ====
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 ^                                                                                                                                           ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ^ ^                                                                                                                                           ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ^
-| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?400}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |+| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?100}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |
 | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]]                                  | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.                                                                                                                                        | | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]]                                  | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.                                                                                                                                        |
 | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]]                            | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]]                            | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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 ^                                                                                                                                                                                                         ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ^ ^                                                                                                                                                                                                         ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ^
-| [[https://www.amazon.com/Space-Time-Algebra-David-Hestenes/dp/3319184121|{{:ga:space-time_algebra-hestenes.jpg?650}}]] | **Space-Time Algebra, 2nd Ed (2015)**\\ //David Hestenes//\\ This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |+| [[https://www.amazon.com/Space-Time-Algebra-David-Hestenes/dp/3319184121|{{:ga:space-time_algebra-hestenes.jpg?100}}]] | **Space-Time Algebra, 2nd Ed (2015)**\\ //David Hestenes//\\ This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
 | [[https://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148|{{:ga:new_foundations_for_classical_mechanics-hestenes.jpg?100}}]] | **New Foundations for Classical Mechanics, 2nd Ed (1999)**\\ //David Hestenes//\\ This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels.                                                                                                                                                                                                                                                                                 | | [[https://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148|{{:ga:new_foundations_for_classical_mechanics-hestenes.jpg?100}}]] | **New Foundations for Classical Mechanics, 2nd Ed (1999)**\\ //David Hestenes//\\ This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels.                                                                                                                                                                                                                                                                                 |
 | [[https://www.amazon.com/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616|{{:ga:clifford_algebra_to_geometric_calculus-hestenes_sobczyk.jpg?100}}]] | **Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (1984)**\\ //David Hestenes, Garret Sobczyk//\\ Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.                                                            | | [[https://www.amazon.com/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616|{{:ga:clifford_algebra_to_geometric_calculus-hestenes_sobczyk.jpg?100}}]] | **Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (1984)**\\ //David Hestenes, Garret Sobczyk//\\ Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.                                                            |
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 | [[https://www.amazon.com/dp/331990664X|{{:ga:a_geometric_algebra_invitation-lavor_xambo.jpg?100}}]] | **A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry (2018)**\\ //Carlile Lavor, Sebastià Xambó-Descamps//\\ This book offers a gentle introduction to key elements of Geometric Algebra, along with their applications in Physics, Robotics and Molecular Geometry. Major applications covered are the physics of space-time, including Maxwell electromagnetism and the Dirac equation; robotics, including formulations for the forward and inverse kinematics and an overview of the singularity problem for serial robots; and molecular geometry, with 3D-protein structure calculations using NMR data. The book is primarily intended for graduate students and advanced undergraduates in related fields, but can also benefit professionals in search of a pedagogical presentation of these subjects.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | | [[https://www.amazon.com/dp/331990664X|{{:ga:a_geometric_algebra_invitation-lavor_xambo.jpg?100}}]] | **A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry (2018)**\\ //Carlile Lavor, Sebastià Xambó-Descamps//\\ This book offers a gentle introduction to key elements of Geometric Algebra, along with their applications in Physics, Robotics and Molecular Geometry. Major applications covered are the physics of space-time, including Maxwell electromagnetism and the Dirac equation; robotics, including formulations for the forward and inverse kinematics and an overview of the singularity problem for serial robots; and molecular geometry, with 3D-protein structure calculations using NMR data. The book is primarily intended for graduate students and advanced undergraduates in related fields, but can also benefit professionals in search of a pedagogical presentation of these subjects.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
 | [[https://www.amazon.com/dp/3319748289|{{:ga:geometric-algebra-applications_vol_i-bayro.jpg?100}}]] | **Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing (2018)**\\ //Eduardo Bayro-Corrochano//\\ The goal is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | | [[https://www.amazon.com/dp/3319748289|{{:ga:geometric-algebra-applications_vol_i-bayro.jpg?100}}]] | **Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing (2018)**\\ //Eduardo Bayro-Corrochano//\\ The goal is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 +| [[https://www.amazon.com/dp/3030349764|{{:ga:geometric-algebra-applications_vol_ii-bayro.jpg?100}}]] | **Geometric Algebra Applications Vol. II: Robot Modelling and Control (2020)**\\ //Eduardo Bayro-Corrochano//\\ This book presents a unified mathematical treatment of diverse problems in the general domain of robotics and associated fields using Clifford or geometric algebra. By addressing a wide spectrum of problems in a common language, it offers both fresh insights and new solutions that are useful to scientists and engineers working in areas related with robotics. It introduces non-specialists to Clifford and geometric algebra, and provides examples to help readers learn how to compute using geometric entities and geometric formulations. It also includes an in-depth study of applications of Lie group theory, Lie algebra, spinors and versors and the algebra of incidence using the universal geometric algebra generated by reciprocal null cones. Featuring a detailed study of kinematics, differential kinematics and dynamics using geometric algebra, the book also develops Euler Lagrange and Hamiltonians equations for dynamics using conformal geometric algebra, and the recursive Newton-Euler using screw theory in the motor algebra framework. Further, it comprehensively explores robot modeling and nonlinear controllers, and discusses several applications in computer vision, graphics, neurocomputing, quantum computing, robotics and control engineering using the geometric algebra framework.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
 | [[https://www.amazon.com/dp/1498748384|{{:ga:introduction_to_geometric_algebra_computing-hildenbrand.jpg?100}}]] | **Introduction to Geometric Algebra Computing (Computer Vision) (2018)**\\ //Dietmar Hildenbrand//\\ This book is intended to give a rapid introduction to computing with Geometric Algebra and its power for geometric modeling. From the geometric objects point of view, it focuses on the most basic ones, namely points, lines and circles. This algebra is called Compass Ruler Algebra, since it is comparable to working with a compass and ruler. The book explores how to compute with these geometric objects, and their geometric operations and transformations, in a very intuitive way. The book follows a top-down approach, and while it focuses on 2D, it is also easily expandable to 3D computations. Algebra in engineering applications such as computer graphics, computer vision and robotics are also covered.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | | [[https://www.amazon.com/dp/1498748384|{{:ga:introduction_to_geometric_algebra_computing-hildenbrand.jpg?100}}]] | **Introduction to Geometric Algebra Computing (Computer Vision) (2018)**\\ //Dietmar Hildenbrand//\\ This book is intended to give a rapid introduction to computing with Geometric Algebra and its power for geometric modeling. From the geometric objects point of view, it focuses on the most basic ones, namely points, lines and circles. This algebra is called Compass Ruler Algebra, since it is comparable to working with a compass and ruler. The book explores how to compute with these geometric objects, and their geometric operations and transformations, in a very intuitive way. The book follows a top-down approach, and while it focuses on 2D, it is also easily expandable to 3D computations. Algebra in engineering applications such as computer graphics, computer vision and robotics are also covered.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 | [[https://www.amazon.com/dp/3319946366|{{:ga:imaginary_mathematics_for_computer_science-vince.jpg?100}}]] | **Imaginary Mathematics for Computer Science (2018)**\\ //John Vince//\\ The imaginary unit //i// has been used by mathematicians for nearly five-hundred years, during which time its physical meaning has been a constant challenge. Unfortunately, Rene Descartes referred to it as "imaginary", and the use of the term "complex number" compounded the unnecessary mystery associated with this amazing object. Today, //i// has found its way into virtually every branch of mathematics, and is widely employed in physics and science, from solving problems in electrical engineering to quantum field theory. John Vince describes the evolution of the imaginary unit from the roots of quadratic and cubic equations, Hamilton's quaternions, Cayley's octonions, to Grassmann's geometric algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | | [[https://www.amazon.com/dp/3319946366|{{:ga:imaginary_mathematics_for_computer_science-vince.jpg?100}}]] | **Imaginary Mathematics for Computer Science (2018)**\\ //John Vince//\\ The imaginary unit //i// has been used by mathematicians for nearly five-hundred years, during which time its physical meaning has been a constant challenge. Unfortunately, Rene Descartes referred to it as "imaginary", and the use of the term "complex number" compounded the unnecessary mystery associated with this amazing object. Today, //i// has found its way into virtually every branch of mathematics, and is widely employed in physics and science, from solving problems in electrical engineering to quantum field theory. John Vince describes the evolution of the imaginary unit from the roots of quadratic and cubic equations, Hamilton's quaternions, Cayley's octonions, to Grassmann's geometric algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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 | [[https://www.amazon.com/Advanced-Calculus-Fundamentals-Carlos-Polanco/dp/9811415072|{{:ga:advanced_calculus-polanco.jpg?100}}]] | **Advanced Calculus - Fundamentals of Mathematics (2019)**\\ // Carlos Polanco//\\ Vector calculus is an essential mathematical tool for performing mathematical analysis of physical and natural phenomena. It is employed in advanced applications in the field of engineering and computer simulations.This textbook covers the fundamental requirements of vector calculus in curricula for college students in mathematics and engineering programs. Chapters start from the basics of vector algebra, real valued functions, different forms of integrals, geometric algebra and the various theorems relevant to vector calculus and differential forms.Readers will find a concise and clear study of vector calculus, along with several examples, exercises, and a case study in each chapter.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | | [[https://www.amazon.com/Advanced-Calculus-Fundamentals-Carlos-Polanco/dp/9811415072|{{:ga:advanced_calculus-polanco.jpg?100}}]] | **Advanced Calculus - Fundamentals of Mathematics (2019)**\\ // Carlos Polanco//\\ Vector calculus is an essential mathematical tool for performing mathematical analysis of physical and natural phenomena. It is employed in advanced applications in the field of engineering and computer simulations.This textbook covers the fundamental requirements of vector calculus in curricula for college students in mathematics and engineering programs. Chapters start from the basics of vector algebra, real valued functions, different forms of integrals, geometric algebra and the various theorems relevant to vector calculus and differential forms.Readers will find a concise and clear study of vector calculus, along with several examples, exercises, and a case study in each chapter.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 | [[https://www.amazon.com/Geometric-Multiplication-Vectors-Introduction-Mathematics/dp/3030017559|{{:ga:geometric_multiplication_of_vectors-josipovic.jpg?100}}]] | **Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics (2019)**\\ // Miroslav Josipović//\\ Enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra? What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics? What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science? In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics? Such a mathematical language exists and it is called geometric algebra. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | | [[https://www.amazon.com/Geometric-Multiplication-Vectors-Introduction-Mathematics/dp/3030017559|{{:ga:geometric_multiplication_of_vectors-josipovic.jpg?100}}]] | **Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics (2019)**\\ // Miroslav Josipović//\\ Enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra? What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics? What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science? In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics? Such a mathematical language exists and it is called geometric algebra. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 +| [[https://www.amazon.com/dp/1704596629|{{:ga:matrix-gateway-to-geometric-algebra_sobczyk.jpg?100}}]] | **Matrix Gateway to Geometric Algebra, Spacetime and Spinors (2019)**\\ // Garret Sobczyk//\\ Geometric algebra has been presented in many different guises since its invention by William Kingdon Clifford shortly before his death in 1879. In this book we fully integrate the ideas of geometric algebra directly into the fabric of matrix linear algebra. A geometric matrix is a real or complex matrix which is identified with a unique geometric number. The matrix product of two geometric matrices is just the product of the corresponding geometric numbers. Any equation can be either interpreted as a matrix equation or an equation in geometric algebra, thus fully unifying the two languages. The first 6 chapters provide an introduction to geometric algebra, and the classification of all such algebras. The last 3 chapters explore more advanced topics in the application of geometric algebras to Pauli and Dirac spinors, special relativity, Maxwell’s equations, quaternions, split quaternions, and group manifolds. They are included to highlight the great variety of topics that are imbued with new geometric insight when expressed in geometric algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 | [[https://www.amazon.com/Clifford-Algebras-Zeons-Geometry-Combinatorics/dp/9811202575|{{:ga:clifford_algebras_and_zeons-staples.jpg?100}}]] | **Clifford Algebras And Zeons: Geometry to Combinatorics and Beyond (2020)**\\ // George Stacey Staples//\\ Clifford algebras have many well-known applications in physics, engineering, and computer graphics. Zeon algebras are subalgebras of Clifford algebras whose combinatorial properties lend them to graph-theoretic applications such as enumerating minimal cost paths in dynamic networks. This book provides a foundational working knowledge of zeon algebras, their properties, and their potential applications in an increasingly technological world. As the first textbook to explore algebraic and combinatorial properties of zeon algebras in depth, it is suitable for interdisciplinary study in analysis, algebra, and combinatorics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | | [[https://www.amazon.com/Clifford-Algebras-Zeons-Geometry-Combinatorics/dp/9811202575|{{:ga:clifford_algebras_and_zeons-staples.jpg?100}}]] | **Clifford Algebras And Zeons: Geometry to Combinatorics and Beyond (2020)**\\ // George Stacey Staples//\\ Clifford algebras have many well-known applications in physics, engineering, and computer graphics. Zeon algebras are subalgebras of Clifford algebras whose combinatorial properties lend them to graph-theoretic applications such as enumerating minimal cost paths in dynamic networks. This book provides a foundational working knowledge of zeon algebras, their properties, and their potential applications in an increasingly technological world. As the first textbook to explore algebraic and combinatorial properties of zeon algebras in depth, it is suitable for interdisciplinary study in analysis, algebra, and combinatorics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
  
geometric_algebra.1595179385.txt.gz · Last modified: 2020/07/19 17:23 by pbk

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