Geometric Algebra with Applications in Science and Engineering

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Eduardo Bayro Corrochano, Garret Sobczyk
Springer Science & Business Media, Apr 20, 2001 - Mathematics - 592 pages
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The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in 1878 as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before. Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. The approach to Clifford algebra adopted in most of the ar ticles here was pioneered in the 1960s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math ematics and physics. Sobczyk first learned about the power of geometric algebra in classes in electrodynamics and relativity taught by Hestenes at Arizona State University from 1966 to 1967. He still vividly remembers a feeling of disbelief that the fundamental geometric product of vectors could have been left out of his undergraduate mathematics education. Geometric algebra provides a rich, general mathematical framework for the develop ment of multilinear algebra, projective and affine geometry, calculus on a manifold, the representation of Lie groups and Lie algebras, the use of the horosphere and many other areas. This book is addressed to a broad audience of applied mathematicians, physicists, computer scientists, and engineers.
 

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Contents

A New Algebraic Framework
3
Universal Geometric Algebra
18
Realizations of the Conformal Group
42
Hyperbolic Geometry
61
Geometric Reasoning With Geometric Algebra
89
Automated Theorem Proving
110
The Geometry Algebra of Computer Vision
123
Using Geometric Algebra for Optical Motion Capture
147
Neural Networks
317
Image Analysis Using Quaternion Wavelets
326
Boundary Collisions as Geometric Wave
349
Modern Geometric Calculations in Crystallography
371
Quaternion Optimization Problems in Engineering
387
Clifford Algebras in Electrical Engineering
413
Applications of Geometric Algebra in Physics and Links
430
Clifford Algebras as Projections of Group Algebras
461

An Application
170
Projective Reconstruction of Shape and Motion Using
190
Robot Kinematics and Flags
211
The Clifford Algebra and the Optimization of Robot
235
Applications of Lie Algebras and the Algebra of Incidence
252
Quantum and Neural Computing
279
Geometric Feedforward Neural Networks and Support Mul
309
Counterexamples for Validation and Discovering of
477
A Geometric Algebra Learning
491
Helmstetter Formula and Rigid Motions with CLIFFORD
512
References
535
Index
583
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