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's most recent, original development of Grassmann-Cayley algebra and geometric algebra and their applications in automated reasoning of classical geometries. It includes 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. Read more... Projective Space, Bracket Algebra and Grassmann-Cayley Algebra; Projective Incidence Geometry with Cayley Bracket Algebra; Projective Conic Geometry and Binary Grassmann-Cayley Algebra; Clifford Algebra and Geometric Algebra; Conformal Geometric Algebra for Classical Geometries; Null Bracket Algebra and Euclidean Geometry
Author(s): Hongbo Li
Publisher: World Scientific Publishing Company
Year: 2008
Language: English
Pages: 533
Contents......Page 12
Foreword......Page 8
Preface......Page 10
1.1 Leibniz's dream......Page 16
1.2 Development of geometric algebras......Page 19
1.3 Conformal geometric algebra......Page 25
1.4 Geometric computing with invariant algebras......Page 27
1.5 From basic invariants to advanced invariants......Page 30
1.6 Geometric reasoning with advanced invariant algebras......Page 33
1.7 Highlights of the chapters......Page 36
2.1 Projective space and classical invariants......Page 40
2.2 Brackets from the symbolic point of view......Page 47
2.3 Covariants, duality and Grassmann-Cayley algebra......Page 52
2.4 Grassmann coalgebra......Page 63
2.5.1 Basic Cayley expansions......Page 71
2.5.2 Cayley expansion theory......Page 74
2.5.3 General Cayley expansions......Page 83
2.6 Grassmann factorization......Page 85
2.7 Advanced invariants and Cayley bracket algebra......Page 96
3.1 Symbolic methods for projective incidence geometry......Page 104
3.2.1 Factorization based on GP relations......Page 111
3.2.2 Factorization based on collinearity constraints......Page 112
3.2.3 Factorization based on concurrency constraints......Page 116
3.3 Contraction techniques in bracket computing......Page 118
3.3.1 Contraction......Page 119
3.3.2 Level contraction......Page 120
3.3.3 Strong contraction......Page 122
3.4.1 Exact division by brackets without common vectors......Page 125
3.4.2 Pseudodivision by brackets with common vectors......Page 128
3.5 Rational invariants......Page 131
3.5.1 Antisymmetrization of rational invariants......Page 132
3.5.2 Symmetrization of rational invariants......Page 138
3.6 Automated theorem proving......Page 141
3.6.1 Construction sequence and elimination sequence......Page 143
3.6.2 Geometric constructions and nondegeneracy conditions......Page 146
3.6.3 Theorem proving algorithm and practice......Page 148
3.7 Erdos' consistent 5-tuples......Page 153
3.7.1 Derivation of the fundamental equations......Page 154
3.7.2 Proof of Theorem 3.40......Page 159
3.7.3 Proof of Theorem 3.39......Page 161
4.1 Conics with bracket algebra......Page 166
4.1.1 Conics determined by points......Page 167
4.1.2 Conics determined by tangents and points......Page 174
4.2 Bracket-oriented representation......Page 180
4.2.1 Representations of geometric constructions......Page 181
4.2.2 Representations of geometric conclusions......Page 189
4.3.1 Conic transformation......Page 193
4.3.2 Pseudoconic transformation......Page 196
4.3.3 Conic contraction......Page 199
4.4.1 Bracket unification......Page 200
4.4.2 Conic Cayley factorization......Page 202
4.5 Automated theorem proving......Page 208
4.5.1 Almost incidence geometry......Page 211
4.5.2 Tangency and polarity......Page 214
4.5.3 Intersection......Page 218
4.6 Conics with quadratic Grassmann-Cayley algebra......Page 223
4.6.1 Quadratic Grassmann space and quadratic bracket algebra......Page 224
4.6.2 Extension and Intersection......Page 229
5.1.1 Inner-product space......Page 234
5.1.2 Inner-product Grassmann algebra......Page 242
5.1.3 Algebras of basic invariants and advanced invariants......Page 248
5.2 Clifford algebra......Page 252
5.3 Representations of Clifford algebras......Page 259
5.3.1 Clifford numbers......Page 262
5.3.2 Matrix-formed Clifford algebras......Page 265
5.3.3 Groups in Clifford algebra......Page 268
5.4.1 Expansion of the geometric product of vectors......Page 270
5.4.2 Expansion of square bracket......Page 274
5.4.3 Expansion of the geometric product of blades......Page 279
6.1 Major techniques in Geometric Algebra......Page 288
6.1.1 Symmetry......Page 293
6.1.2 Commutation......Page 295
6.1.3 Ungrading......Page 300
6.2 Versor compression......Page 304
6.2.1 4-tuple compression......Page 306
6.2.2 5-tuple compression......Page 310
6.2.3 m-tuple compression......Page 315
6.3.1 Almost null space......Page 317
6.3.2 Parabolic rotors......Page 320
6.3.3 Hyperbolic rotors......Page 323
6.3.4 Maximal grade conjectures......Page 330
6.4 Clifford coalgebra, Clifford summation and factorization......Page 333
6.4.1 One Clifford monomial......Page 335
6.4.2 Two Clifford monomials......Page 336
6.4.3 Three Clifford monomials......Page 339
6.4.4 Clifford coproduct of blades......Page 342
6.5 Clifford bracket algebra......Page 347
7.1.1 Affne space and affine Grassmann-Cayley algebra......Page 354
7.1.2 The Cartesian model of Euclidean space......Page 359
7.2 The conformal model and the homogeneous model......Page 361
7.2.1 The conformal model......Page 362
7.2.2 Vectors of different signatures......Page 365
7.2.3 The homogeneous model......Page 368
7.3 Positive-vector representations of spheres and hyperplanes......Page 369
7.3.1 Pencils of spheres and hyperplanes......Page 370
7.3.2 Positive-vector representation......Page 372
7.4 Conformal Grassmann-Cayley algebra......Page 377
7.4.1 Geometry of Minkowski blades......Page 378
7.4.2 Inner product of Minkowski blades......Page 385
7.4.3 Meet product of Minkowski blades......Page 390
7.5 The Lie model of oriented spheres and hyperplanes......Page 400
7.5.1 Inner product of Lie spheres......Page 402
7.5.2 Lie pencils, positive vectors and negative vectors......Page 406
7.6.1 1D contact problem......Page 414
7.6.2 2D contact problem......Page 415
7.6.3 nD contact problem......Page 422
8.1 The geometry of positive monomials......Page 426
8.1.1 Versors for conformal transformations......Page 427
8.1.2 Geometric product of Minkowski blades......Page 434
8.2 Cayley transform and exterior exponential......Page 440
8.3 Twisted Vahlen matrices and Vahlen matrices......Page 450
8.4 Affne geometry with dual Clifford algebra......Page 457
8.5.1 The classical model of spherical geometry......Page 463
8.5.2 The conformal model of spherical geometry......Page 465
8.6 Hyperbolic geometry and its conformal model......Page 467
8.6.1 Poincar e's hyperboloid model of hyperbolic geometry......Page 468
8.6.2 The conformal model of double-hyperbolic geometry......Page 474
8.6.3 Poincar e's disk model and half-space model......Page 475
8.7 Unified algebraic framework for classical geometries......Page 477
A.1 Cayley expansions of pII......Page 484
A.2 Cayley expansions of pIII......Page 485
A.3 Cayley expansions of pIV......Page 491
A.4 Cayley expansions of qI ; qII and qIII......Page 505
A.5 Cayley expansions of rI and rII......Page 507
Bibliography......Page 510
Index......Page 520
