| Both sides previous revisionPrevious revision | Next revisionBoth sides next revision |
| geometric_algebra [2020/07/19 17:23] – [Articles] pbk | geometric_algebra [2020/07/19 17:34] – [Articles] pbk |
|---|
| 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. |
| |
| |
| * [[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// |
| 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. |
| |
| |
| * [[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. |
| |
| * [[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// |
| |
| * [[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// |
| |
| * [[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. |
| |
| * [[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.// |
| |
| * [[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// |
| |
| * [[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// |
| |
| * [[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// |
| |
| * [[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// |
| * [[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// |
| |
| * [[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// |
