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geometric_algebra [2020/07/10 17:04] – [Computing frameworks] pbk | geometric_algebra [2020/09/07 19:40] – [Computing frameworks] pbk |
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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]] |
* [[https://en.wikipedia.org/wiki/Multivector|Multivector]] | * [[https://en.wikipedia.org/wiki/Multivector|Multivector]] |
* [[https://en.wikipedia.org/wiki/Paravector|Paravector]] | * [[https://en.wikipedia.org/wiki/Paravector|Paravector]] |
| * [[https://en.wikipedia.org/wiki/Outermorphism|Outermorphism]] |
* [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]] | * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]] |
* [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]] | * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]] |
* [[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/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// |
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. |
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* [[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. |
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* [[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// |
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. |
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* [[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 a new venue of research in control of manipulators and robot legs that involves the use of geometric primitives, such as lines, circles, planes, spheres. | Controlling the pose of a manipulator involves finding the correct configuration of the robot’s elements to move the end effector to a desired position and orientation. In order to find the geometric relationships between the elements of a robot manipulator, it is necessary to define the kinematics of the robot. We present a synthesis of the kinematical model of the pose for this type of robot using the conformal geometric algebra framework. In addition, two controllers are developed, one for the position tracking problem and another for the orientation tracking problem, both using an error feedback controller. The stability analysis is carried out for both controllers, and their application to a 6-DOF serial manipulator and the legs of a biped robot are presented. By proposing the error feedback and Lyapunov functions in terms of geometric algebra, we are opening 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. |
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* [[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. |
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* [[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// |
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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 a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions. | This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions. |
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* [[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. |
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* [[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. |
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* [[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. |
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* [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano// | * [[http://downloads.hindawi.com/journals/abb/2007/502679.pdf|Surface Approximation using Growing Self-Organizing Nets and Gradient Information]] (2007) - //Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano// |
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* [[http://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IFAC_ICINCO_2007/ICINCO%202007/Area%202%20-%20Robotics%20and%20Automation/Volume%202/Short%20Papers/C2_496_Bayro-Corrochano.pdf|Geometric Advanced Techniques for Robot Grasping using Stereoscopic Vision]] (2007) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano// | * [[http://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IFAC_ICINCO_2007/ICINCO%202007/Area%202%20-%20Robotics%20and%20Automation/Volume%202/Short%20Papers/C2_496_Bayro-Corrochano.pdf|Geometric Advanced Techniques for Robot Grasping using Stereoscopic Vision]] (2007) - //Julio Zamora-Esquivel, Eduardo Bayro-Corrochano// |
In this paper the authors propose geometric techniques to deal with the problem of grasping objects relying on their mathematical models. For that we use the geometric algebra framework to formulate the kinematics of a three finger robotic hand. Our main objective is by formulating a kinematic control law to close the loop between perception and actions. This allows us to perform a smooth visually guided object grasping action. | In this paper the authors propose geometric techniques to deal with the problem of grasping objects relying on their mathematical models. For that we use the geometric algebra framework to formulate the kinematics of a three finger robotic hand. Our main objective is by formulating a kinematic control law to close the loop between perception and actions. This allows us to perform a smooth visually guided object grasping action. |
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* [[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// |
* [[http://cs.unitbv.ro/~acami/what_kc_can_do.pdf|What the Kähler Calculus can do that other calculi cannot]] (2016) - //Jose G. Vargas// | * [[http://cs.unitbv.ro/~acami/what_kc_can_do.pdf|What the Kähler Calculus can do that other calculi cannot]] (2016) - //Jose G. Vargas// |
Underlied by Clifford algebra of differential forms --- like tangent Clifford algebra underlies the geometric calculus --- it brings about a fresh new view of quantum mechanics. This view arises, almost without effort, from the equation which is in Kähler Calculus what the Dirac equation is in traditional quantum mechanics. | Underlied by Clifford algebra of differential forms --- like tangent Clifford algebra underlies the geometric calculus --- it brings about a fresh new view of quantum mechanics. This view arises, almost without effort, from the equation which is in Kähler Calculus what the Dirac equation is in traditional quantum mechanics. |
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| * [[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. |
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| * [[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. |
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* [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie// | * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie// |
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* [[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. |
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* [[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// |
* [[https://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez// | * [[https://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez// |
Robust methods for finding the best rotation aligning two sets of corresponding vectors are formulated in the linear algebra framework, using tools like the SVD for polar decomposition or QR for finding eigenvectors. Those are well established numerical algorithms which on the other hand are iterative and computationally expensive. Recently, closed form solutions has been proposed in the quaternion’s framework, those methods are fast but they have singularities i.e., they completely fail on certain input data. In this paper we propose a robust attitude estimator based on a formulation of the problem in Geometric Algebra. We find the optimal eigen-quaternion in closed form with high accuracy and with competitive performance respect to the fastest methods reported in literature. | Robust methods for finding the best rotation aligning two sets of corresponding vectors are formulated in the linear algebra framework, using tools like the SVD for polar decomposition or QR for finding eigenvectors. Those are well established numerical algorithms which on the other hand are iterative and computationally expensive. Recently, closed form solutions has been proposed in the quaternion’s framework, those methods are fast but they have singularities i.e., they completely fail on certain input data. In this paper we propose a robust attitude estimator based on a formulation of the problem in Geometric Algebra. We find the optimal eigen-quaternion in closed form with high accuracy and with competitive performance respect to the fastest methods reported in literature. |
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| * [[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. |
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* [[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// |
* [[https://arxiv.org/pdf/2002.05993|Projective Geometric Algebra as a Subalgebra of Conformal Geometric Algebra]] (2020) - //Jaroslav Hrdina, Ales Navrat, Petr Vasik, Dietmar Hildenbrand// | * [[https://arxiv.org/pdf/2002.05993|Projective Geometric Algebra as a Subalgebra of Conformal Geometric Algebra]] (2020) - //Jaroslav Hrdina, Ales Navrat, Petr Vasik, Dietmar Hildenbrand// |
First we introduce briefly the frameworks of CGA and PGA for doing Euclidean geometry and we summarise basic formulas. In the next section, we show that there are actually two naturally related copies of PGA in CGA. After an identification of the two copies, the duality in PGA is obtained in terms of CGA operations. This implies directly the correspondence between flat objects and versors for Euclidean transformations in CGA and the objects and versors in PGA. | First we introduce briefly the frameworks of CGA and PGA for doing Euclidean geometry and we summarise basic formulas. In the next section, we show that there are actually two naturally related copies of PGA in CGA. After an identification of the two copies, the duality in PGA is obtained in terms of CGA operations. This implies directly the correspondence between flat objects and versors for Euclidean transformations in CGA and the objects and versors in PGA. |
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| * [[https://link.springer.com/content/pdf/10.1007/s00006-020-1046-0.pdf|A 1d Up Approach to Conformal Geometric Algebra]] (2020) - //Anthony N. Lasenby// |
| We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell's Theorem in quantum mechanics, and its relation to the sphere S^7 and the exceptional group E_8, and proposed a new associative version of the division algebra normally thought to require the octonians. |
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* [[https://arxiv.org/pdf/2004.06655|Dimensional scaffolding of electromagnetism using geometric algebra]] (2020) - //Xabier Prado Orbán, Jorge Mira// | * [[https://arxiv.org/pdf/2004.06655|Dimensional scaffolding of electromagnetism using geometric algebra]] (2020) - //Xabier Prado Orbán, Jorge Mira// |
Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. (...) Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism. | Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. (...) Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism. |
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| * [[https://www.ram-lab.com/papers/2020/wu2020iet.pdf|A Linear Geometric Algebra Rotor Estimator for Efficient Mesh Deformation]] (2020) - //Jin Wu, Mauricio Lopez, Ming Liu, Yilong Zhu// |
| We solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. We show its efficiency in two applications: the As-Rigid-As-Possible (ARAP) Surface Modeling and the more Smooth Rotation enhanced As-Rigid-As-Possible (SR-ARAP) mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics. |
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* [[http://publikacio.uni-eszterhazy.hu/5004/1/AMI_online_1059.pdf|Optimized line and line segment clipping in E2 and Geometric Algebra]] (2020) - //Vaclav Skala// | * [[http://publikacio.uni-eszterhazy.hu/5004/1/AMI_online_1059.pdf|Optimized line and line segment clipping in E2 and Geometric Algebra]] (2020) - //Vaclav Skala// |