Invariant algebras and geometric reasoning
The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics — among them, Grassmann–Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other...
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| Médium: | E-kniha Kniha |
| Jazyk: | English |
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New Jersey
World Scientific Publishing Co. Pte. Ltd
2008
World Scientific World Scientific Publishing Company WORLD SCIENTIFIC WSPC |
| Vydanie: | 1 |
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| ISBN: | 9789812708083, 9812708081, 9812770119, 9789812770110 |
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| Abstract | The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics — among them, Grassmann–Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries. |
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| AbstractList | The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics — among them, Grassmann–Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries. The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics - among them, Grassmann-Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries.This book contains the author and his collaborators' most recent, original development of Grassmann-Cayley algebra and Geometric Algebra and their applications in automated reasoning of classical geometries. It includes two of the three advanced invariant algebras - Cayley bracket algebra, conformal geometric algebra, and null bracket algebra - for highly efficient geometric computing. They form the theory of advanced invariants, and capture the intrinsic beauty of geometric languages and geometric computing. Apart from their applications in discrete and computational geometry, the new languages are currently being used in computer vision, graphics and robotics by many researchers worldwide.Sample Chapter(s)Chapter 1: Introduction (252 KB)Contents:Projective Space, Bracket Algebra and Grassmann-Cayley AlgebraProjective Incidence Geometry with Cayley Bracket AlgebraProjective Conic Geometry with Bracket Algebra and Quadratic Grassmann-Cayley AlgebraInner-product Bracket Algebra and Clifford AlgebraGeometric AlgebraEuclidean Geometry and Conformal Grassmann-Cayley AlgebraConformal Clifford Algebra and Classical GeometriesReadership: Graduate students in discrete and computational geometry, and computer mathematics; mathematicians and computer scientists. The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics - among them, Grassmann-Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries. This book contains the author and his collaborators' most recent, original development of Grassmann-Cayley algebra and Geometric Algebra and their applications in automated reasoning of classical geometries. It includes two of the three advanced invariant algebras - Cayley bracket algebra, conformal geometric algebra, and null bracket algebra - for highly efficient geometric computing. They form the theory of advanced invariants, and capture the intrinsic beauty of geometric languages and geometric computing. Apart from their applications in discrete and computational geometry, the new languages are currently being used in computer vision, graphics and robotics by many researchers worldwide. |
| Author | Li, Hongbo |
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| Keywords | Bracket Algebra Invariant Theory Automated Theorem Proving Geometric Algebra Geometric Invariance Computational Geometry Discrete Geometry Projective Geometry Geometric Reasoning Grassmann-Cayley Algebra |
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| Notes | Includes bibliographical references and index |
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| Snippet | The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics —... The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics -... |
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| SubjectTerms | Artificial Intelligence (Machine Learning, Neural Networks, Fuzzy Logic) Clifford algebras Conformal geometry Invariants Pure Mathematics SCIENCE Symmetry (Mathematics) |
| SubjectTermsDisplay | Clifford algebras Invariants Symmetry (Mathematics) |
| TableOfContents | Invariant algebras and geometric reasoning -- Foreword -- Preface -- Contents -- Chapter 1: Introduction -- Chapter 2: Projective Space, Bracket Algebra and Grassmann-Cayley Algebra -- Chapter 3: Projective Incidence Geometry with Cayley Bracket Algebra -- Chapter 4: Projective Conic Geometry with Bracket Algebra and Quadratic Grassmann-Cayley Algebra -- Chapter 5: Inner-product Bracket Algebra and Clifford Algebra -- Chapter 6: Geometric Algebra -- Chapter 7: Euclidean Geometry and Conformal Grassmann-Cayley Algebra -- Chapter 8: Conformal Clifford Algebra and Classical Geometries -- Appendix A: Cayley Expansion Theory for 2D and 3D Projective Geometries -- Bibliography -- Index Intro -- Contents -- Foreword -- Preface -- 1. Introduction -- 1.1 Leibniz's dream -- 1.2 Development of geometric algebras -- 1.3 Conformal geometric algebra -- 1.4 Geometric computing with invariant algebras -- 1.5 From basic invariants to advanced invariants -- 1.6 Geometric reasoning with advanced invariant algebras -- 1.7 Highlights of the chapters -- 2. Projective Space, Bracket Algebra and Grassmann-Cayley Algebra -- 2.1 Projective space and classical invariants -- 2.2 Brackets from the symbolic point of view -- 2.3 Covariants, duality and Grassmann-Cayley algebra -- 2.4 Grassmann coalgebra -- 2.5 Cayley expansion -- 2.5.1 Basic Cayley expansions -- 2.5.2 Cayley expansion theory -- 2.5.3 General Cayley expansions -- 2.6 Grassmann factorization -- 2.7 Advanced invariants and Cayley bracket algebra -- 3. Projective Incidence Geometry with Cayley Bracket Algebra -- 3.1 Symbolic methods for projective incidence geometry -- 3.2 Factorization techniques in bracket algebra -- 3.2.1 Factorization based on GP relations -- 3.2.2 Factorization based on collinearity constraints -- 3.2.3 Factorization based on concurrency constraints -- 3.3 Contraction techniques in bracket computing -- 3.3.1 Contraction -- 3.3.2 Level contraction -- 3.3.3 Strong contraction -- 3.4 Exact division and pseudodivision -- 3.4.1 Exact division by brackets without common vectors -- 3.4.2 Pseudodivision by brackets with common vectors -- 3.5 Rational invariants -- 3.5.1 Antisymmetrization of rational invariants -- 3.5.2 Symmetrization of rational invariants -- 3.6 Automated theorem proving -- 3.6.1 Construction sequence and elimination sequence -- 3.6.2 Geometric constructions and nondegeneracy conditions -- 3.6.3 Theorem proving algorithm and practice -- 3.7 Erdos' consistent 5-tuples -- 3.7.1 Derivation of the fundamental equations -- 3.7.2 Proof of Theorem 3.40 8.6.2 The conformal model of double-hyperbolic geometry -- 8.6.3 Poincar e's disk model and half-space model -- 8.7 Unified algebraic framework for classical geometries -- Appendix A Cayley Expansion Theory for 2D and 3D Projective Geometries -- A.1 Cayley expansions of pII -- A.2 Cayley expansions of pIII -- A.3 Cayley expansions of pIV -- A.4 Cayley expansions of qI -- qII and qIII -- A.5 Cayley expansions of rI and rII -- Bibliography -- Index 3.7.3 Proof of Theorem 3.39 -- 4. Projective Conic Geometry with Bracket Algebra and Quadratic Grassmann-Cayley Algebra -- 4.1 Conics with bracket algebra -- 4.1.1 Conics determined by points -- 4.1.2 Conics determined by tangents and points -- 4.2 Bracket-oriented representation -- 4.2.1 Representations of geometric constructions -- 4.2.2 Representations of geometric conclusions -- 4.3 Simplification techniques in conic computing -- 4.3.1 Conic transformation -- 4.3.2 Pseudoconic transformation -- 4.3.3 Conic contraction -- 4.4 Factorization techniques in conic computing -- 4.4.1 Bracket unification -- 4.4.2 Conic Cayley factorization -- 4.5 Automated theorem proving -- 4.5.1 Almost incidence geometry -- 4.5.2 Tangency and polarity -- 4.5.3 Intersection -- 4.6 Conics with quadratic Grassmann-Cayley algebra -- 4.6.1 Quadratic Grassmann space and quadratic bracket algebra -- 4.6.2 Extension and Intersection -- 5. Inner-product Bracket Algebra and Clifford Algebra -- 5.1 Inner-product bracket algebra -- 5.1.1 Inner-product space -- 5.1.2 Inner-product Grassmann algebra -- 5.1.3 Algebras of basic invariants and advanced invariants -- 5.2 Clifford algebra -- 5.3 Representations of Clifford algebras -- 5.3.1 Clifford numbers -- 5.3.2 Matrix-formed Clifford algebras -- 5.3.3 Groups in Clifford algebra -- 5.4 Clifford expansion theory -- 5.4.1 Expansion of the geometric product of vectors -- 5.4.2 Expansion of square bracket -- 5.4.3 Expansion of the geometric product of blades -- 6. Geometric Algebra -- 6.1 Major techniques in Geometric Algebra -- 6.1.1 Symmetry -- 6.1.2 Commutation -- 6.1.3 Ungrading -- 6.2 Versor compression -- 6.2.1 4-tuple compression -- 6.2.2 5-tuple compression -- 6.2.3 m-tuple compression -- 6.3 Obstructions to versor compression -- 6.3.1 Almost null space -- 6.3.2 Parabolic rotors -- 6.3.3 Hyperbolic rotors 6.3.4 Maximal grade conjectures -- 6.4 Clifford coalgebra, Clifford summation and factorization -- 6.4.1 One Clifford monomial -- 6.4.2 Two Clifford monomials -- 6.4.3 Three Clifford monomials -- 6.4.4 Clifford coproduct of blades -- 6.5 Clifford bracket algebra -- 7. Euclidean Geometry and Conformal Grassmann-Cayley Algebra -- 7.1 Homogeneous coordinates and Cartesian coordinates -- 7.1.1 Affne space and affine Grassmann-Cayley algebra -- 7.1.2 The Cartesian model of Euclidean space -- 7.2 The conformal model and the homogeneous model -- 7.2.1 The conformal model -- 7.2.2 Vectors of different signatures -- 7.2.3 The homogeneous model -- 7.3 Positive-vector representations of spheres and hyperplanes -- 7.3.1 Pencils of spheres and hyperplanes -- 7.3.2 Positive-vector representation -- 7.4 Conformal Grassmann-Cayley algebra -- 7.4.1 Geometry of Minkowski blades -- 7.4.2 Inner product of Minkowski blades -- 7.4.3 Meet product of Minkowski blades -- 7.5 The Lie model of oriented spheres and hyperplanes -- 7.5.1 Inner product of Lie spheres -- 7.5.2 Lie pencils, positive vectors and negative vectors -- 7.6 Apollonian contact problem -- 7.6.1 1D contact problem -- 7.6.2 2D contact problem -- 7.6.3 nD contact problem -- 8. Conformal Clifford Algebra and Classical Geometries -- 8.1 The geometry of positive monomials -- 8.1.1 Versors for conformal transformations -- 8.1.2 Geometric product of Minkowski blades -- 8.2 Cayley transform and exterior exponential -- 8.3 Twisted Vahlen matrices and Vahlen matrices -- 8.4 Affne geometry with dual Clifford algebra -- 8.5 Spherical geometry and its conformal model -- 8.5.1 The classical model of spherical geometry -- 8.5.2 The conformal model of spherical geometry -- 8.6 Hyperbolic geometry and its conformal model -- 8.6.1 Poincar e's hyperboloid model of hyperbolic geometry |
| Title | Invariant algebras and geometric reasoning |
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