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| geometric_algebra [2020/03/22 19:44] – [Conferences] pbk | geometric_algebra [2020/04/08 18:10] – [Historical] pbk |
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| * [[https://www.youtube.com/watch?v=d-4vYtFfet8|Tutorial: Geometric Computing in Computer Graphics using Conformal Geometric Algebra (Japanese)]] - //Kuma Dasu//. | * [[https://www.youtube.com/watch?v=d-4vYtFfet8|Tutorial: Geometric Computing in Computer Graphics using Conformal Geometric Algebra (Japanese)]] - //Kuma Dasu//. |
| * [[https://www.youtube.com/watch?v=ikCIUzX9myY|Joan Lasenby on Applications of Geometric Algebra in Engineering]] - //Y Combinator//. | * [[https://www.youtube.com/watch?v=ikCIUzX9myY|Joan Lasenby on Applications of Geometric Algebra in Engineering]] - //Y Combinator//. |
| * [[https://www.youtube.com/watch?v=syyK6hTWT7U|Let's remove Quaternions from every 3D Engine]] - //Marc ten Bosch//. | * [[https://www.youtube.com/watch?v=Idlv83CxP-8|Let's remove Quaternions from every 3D Engine]] - //Marc ten Bosch//. |
| * [[https://www.youtube.com/watch?v=hbhxRM_YMv0|Overview of Geometric Algebra by Dr. Jack Hanlon]] - //Aaron Murakami//. | * [[https://www.youtube.com/watch?v=hbhxRM_YMv0|Overview of Geometric Algebra]] - //Jack Hanlon//, via //Aaron Murakami//. |
| * [[https://www.youtube.com/watch?v=P2ZxxoS5YD0|Intro to clifford, a python package for geometric algebra]] - //Alex Arsenovic (810 Labs)//. | * [[https://www.youtube.com/watch?v=P2ZxxoS5YD0|Intro to clifford, a python package for geometric algebra]] - //Alex Arsenovic (810 Labs)//. |
| * [[https://www.youtube.com/watch?v=QbYao72-V6U|Gamma Matrices and the Clifford Algebra]] - //Pretty Much Physics//. | * [[https://www.youtube.com/watch?v=QbYao72-V6U|Gamma Matrices and the Clifford Algebra]] - //Pretty Much Physics//. |
| * [[https://www.youtube.com/watch?v=yG8YKw25f6Y|A Brief introduction to Clifford Algebras by Johannes Familton]] - JMM2018 Quaternion Session, //Quaternion Notices//. | * [[https://www.youtube.com/watch?v=yG8YKw25f6Y|JMM2018: A Brief introduction to Clifford Algebras]] - //Johannes Familton//. |
| | * [[https://www.youtube.com/watch?v=eQjDN0JQ6-s|JuliaCon 2019: Geometric algebra in Julia with Grassmann.jl]] - //Michael Reed//. |
| * [[https://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//. |
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| ===== Computing frameworks ===== | ===== Computing frameworks ===== |
| * [[https://arxiv.org/pdf/1411.6502.pdf|Geometric Algebras for Euclidean Geometry]] (2016) - //Charles G. Gunn// | * [[https://arxiv.org/pdf/1411.6502.pdf|Geometric Algebras for Euclidean Geometry]] (2016) - //Charles G. Gunn// |
| The discussion of how to apply geometric algebra to euclidean n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from 19th century mathematics. We then introduce the dual projectivized Clifford algebra P(R∗_n,0,1) (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA. | The discussion of how to apply geometric algebra to euclidean n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from 19th century mathematics. We then introduce the dual projectivized Clifford algebra P(R∗_n,0,1) (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA. |
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| | * [[https://arxiv.org/pdf/1501.06511.pdf|Doing euclidean plane geometry using projective geometric algebra]] (2016) - //Charles G. Gunn// |
| | The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief review of PGA, the article focuses on P(R∗_2,0,1), the PGA for euclidean plane geometry. It first explores the geometric product involving pairs and triples of basic elements (points and lines), establishing a wealth of fundamental metric and non-metric properties. It then applies the algebra to a variety of familiar topics in plane euclidean geometry and shows that it compares favorably with other approaches in regard to completeness, compactness, practicality, and elegance. The seamless integration of euclidean and ideal (aka infinite) elements forms an essential and novel feature of the treatment. |
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| * [[http://www.gaalop.de/wp-content/uploads/CGI_CGA_Paper.pdf|An inclusive Conformal Geometric Algebra GPU animation interpolation and deformation algorithm]] (2016) | * [[http://www.gaalop.de/wp-content/uploads/CGI_CGA_Paper.pdf|An inclusive Conformal Geometric Algebra GPU animation interpolation and deformation algorithm]] (2016) |
| * [[https://arxiv.org/pdf/1902.05478.pdf|A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks]] (2019) - //Fidelis Zanetti de Castro, Marcos Eduardo Valle// | * [[https://arxiv.org/pdf/1902.05478.pdf|A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks]] (2019) - //Fidelis Zanetti de Castro, Marcos Eduardo Valle// |
| In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras. | In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras. |
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| | * [[https://arxiv.org/pdf/1901.05873.pdf|Projective geometric algebra: A new framework for doing euclidean geometry]] (2019) - //Charles G. Gunn// |
| | A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior algebras as sub-algebras simplifies the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity. |
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| ===== Books ===== | ===== Books ===== |
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| | [[https://archive.org/details/geometricalgebra033556mbp|{{:ga:geometric_algebra-artin_ip.jpg?100}}]] | **Geometric Algebra (1957)**\\ //Emil Artin//\\ Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups. | | | [[https://archive.org/details/geometricalgebra033556mbp|{{:ga:geometric_algebra-artin_ip.jpg?100}}]] | **Geometric Algebra (1957)**\\ //Emil Artin//\\ Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups. | |
| | [[https://archive.org/details/cliffordnumberss00ries|{{:ga:clifford_numbers_and_spinors-riesz.jpg?100}}]] | **Clifford Numbers and Spinors (1958)**\\ //Marcel Riesz//\\ The presentation of the theory of Clifford numbers and spinors given here has grown out of thirteen lectures delivered at the Institute for Fluid Dynamics and Applied Mathematics of the University of Maryland. The work is divided into six chapters which, for the convenience of those readers who are only interested in certain parts of the material treated, are largely independent of each other. This arrangement has, of course, certain disadvantages such as repetitions and over-lappings. | | | [[https://archive.org/details/cliffordnumberss00ries|{{:ga:clifford_numbers_and_spinors-riesz.jpg?100}}]] | **Clifford Numbers and Spinors (1958)**\\ //Marcel Riesz//\\ The presentation of the theory of Clifford numbers and spinors given here has grown out of thirteen lectures delivered at the Institute for Fluid Dynamics and Applied Mathematics of the University of Maryland. The work is divided into six chapters which, for the convenience of those readers who are only interested in certain parts of the material treated, are largely independent of each other. This arrangement has, of course, certain disadvantages such as repetitions and over-lappings. | |
| | [[https://archive.org/details/ElementaryDifferentialGeometry|{{:ga:elementary_differential_geometry-oneill.jpg?100}}]] | **Elementary Differential Geometry (1966)**\\ //Barrett O'Neill//\\ The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry. The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation. | | | [[https://archive.org/details/ElementaryDifferentialGeometry|{{:ga:elementary_differential_geometry-oneill.jpg?100}}]] | **Elementary Differential Geometry (1966)**\\ //Barrett O'Neill//\\ The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry. The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation. | |
| | | [[https://www.cambridge.org/core/books/topological-geometry/AAEBEBC695CF4A98242A74EA2C59E212|{{:ga:topological_geometry-porteus.jpg?100}}]] | **Topological Geometry, 2nd Edition (1981)**\\ //Ian R. Porteous//\\ The earlier chapters of this self-contained text provide a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place. In parallel with this is an account, also from first principles, of the elementary theory of topological spaces and of continuous and differentiable maps that leads up to the definitions of smooth manifolds and their tangent spaces and of Lie groups and Lie algebras. The calculus is presented as far as possible in basis free form to emphasize its geometrical flavor and its linear algebra content. In this second edition Dr. Porteous has taken the opportunity to add a chapter on triality which extends earlier work on the Spin groups in the chapter on Clifford algebras. The details include a number of important transitive group actions and a description of one of the exceptional Lie groups, the group G2. | |
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| ==== Modern ==== | ==== Modern ==== |
