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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| Format: | Electronic eBook |
| Language: | English |
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Singarore ; Hackensack, N.J. :
World Scientific,
2008
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| Local Note: | ProQuest Ebook Central |
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| 100 | 1 | |a Li, Hongbo. | |
| 245 | 1 | 0 | |a Invariant algebras and geometric reasoning / |c Hongbo Li. |
| 260 | |a Singarore ; |a Hackensack, N.J. : |b World Scientific, |c ©2008. | ||
| 300 | |a 1 online resource (xiv, 518 pages) : |b illustrations | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
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| 504 | |a Includes bibliographical references (pages 495-504) and index. | ||
| 505 | 0 | |a 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. 2.6. 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. 3.3. Contraction techniques in bracket computing. 3.4. Exact division and pseudodivision. 3.5. Rational invariants. 3.6. Automated theorem proving. 3.7. Erdös' consistent 5-tuples -- 4. Projective conic geometry with bracket algebra and quadratic Grassmann-Cayley algebra. 4.1. Conics with bracket algebra, 4.2. Bracket-oriented representation. 4.3. Simplification techniques in conic computing. 4.4. Factorization techniques in conic computing. 4.5. Automated theorem proving. 4.6. Conics with quadratic Grassmann-Cayley algebra -- 5. Inner-product bracket algebra and Clifford algebra. 5.1. Inner-product bracket algebra. 5.2. Clifford algebra. 5.3. Representations of Clifford algebras. 5.4. Clifford expansion theory -- 6. Geometric algebra. 6.1. Major techniques in geometric algebra. 6.2. Versor compression. 6.3. Obstructions to versor compression. 6.4. Clifford coalgebra, Clifford summation and factorization. 6.5. Clifford bracket algebra -- 7. Euclidean geometry and conformal Grassmann-Cayley algebra. 7.1. Homogeneous coordinates and Cartesian coordinates. 7.2. The conformal model and the homogeneous model. 7.3. Positive-vector representations of spheres and hyperplanes. 7.4. Conformal Grassmann-Cayley algebra. 7.5. The Lie model of oriented spheres and hyperplanes. 7.6. Apollonian contact problem -- 8. Conformal Clifford algebra and classical geometries. 8.1. The geometry of positive monomials. 8.2. Cayley transform and exterior exponential. 8.3. Twisted Vahlen matrices and Vahlen matrices. 8.4. Affine geometry with dual Clifford algebra. 8.5. Spherical geometry and its conformal model. 8.6. Hyperbolic geometry and its conformal model. 8.7. Unifed algebraic framework for classical geometries. | |
| 520 | |a 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. | ||
| 546 | |a English. | ||
| 588 | 0 | |a Print version record. | |
| 590 | |a ProQuest Ebook Central |b Ebook Central Academic Complete | ||
| 650 | 0 | |a Clifford algebras. | |
| 650 | 0 | |a Invariants. | |
| 650 | 0 | |a Symmetry (Mathematics) | |
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