Invariant algebras and geometric reasoning

• Main Author: Li, Hongbo
• Format: eBook Book
• Language: English
• Published: New Jersey World Scientific Publishing Co. Pte. Ltd 2008; World Scientific; World Scientific Publishing Company; WORLD SCIENTIFIC; WSPC
• Edition: 1
• Subjects: Artificial Intelligence (Machine Learning, Neural Networks, Fuzzy Logic); Clifford algebras; Conformal geometry; Invariants; Pure Mathematics; SCIENCE; Symmetry (Mathematics); Bracket Algebra; Invariant Theory; Automated Theorem Proving; Geometric Algebra; Geometric Invariance; Computational Geometry; Discrete Geometry; Projective Geometry; Geometric Reasoning; Grassmann-Cayley Algebra
• ISBN: 9789812708083; 9812708081; 9812770119; 9789812770110
• DOI: 10.1142/6514

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...

Table of Contents:

• 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