Geometric Algebra with Applications in Science and Engineering

✍ Scribed by Prof. David Hestenes (auth.), Dr. Eduardo Bayro Corrochano, Prof. Garret Sobczyk (eds.)

• Publisher: Birkhäuser Basel
• Year: 2001
• Tongue: English
• Leaves: 607
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xxvi
Front Matter....Pages 1-1
Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry....Pages 3-17
Universal Geometric Algebra....Pages 18-41
Realizations of the Conformal Group....Pages 42-60
Hyperbolic Geometry....Pages 61-85
Front Matter....Pages 87-87
Geometric Reasoning With Geometric Algebra....Pages 89-109
Automated Theorem Proving....Pages 110-119
Front Matter....Pages 121-121
The Geometry Algebra of Computer Vision....Pages 123-146
Using Geometric Algebra for Optical Motion Capture....Pages 147-169
Bayesian Inference and Geometric Algebra: An Application to Camera Localization....Pages 170-189
Projective Reconstruction of Shape and Motion Using Invariant Theory....Pages 190-208
Front Matter....Pages 209-209
Robot Kinematics and Flags....Pages 211-234
The Clifford Algebra and the Optimization of Robot Design....Pages 235-251
Applications of Lie Algebras and the Algebra of Incidence....Pages 252-277
Front Matter....Pages 279-279
Geometric Algebra in Quantum Information Processing by Nuclear Magnetic Resonance....Pages 281-308
Geometric Feedforward Neural Networks and Support Multivector Machines....Pages 309-325
Image Analysis Using Quaternion Wavelets....Pages 326-345
Front Matter....Pages 347-347
Objects in Contact: Boundary Collisions as Geometric Wave Propagation....Pages 349-370
Modern Geometric Calculations in Crystallography....Pages 371-386
Quaternion Optimization Problems in Engineering....Pages 387-412
Clifford Algebras in Electrical Engineering....Pages 413-429
Front Matter....Pages 347-347
Applications of Geometric Algebra in Physics and Links With Engineering....Pages 430-457
Front Matter....Pages 459-459
Clifford Algebras as Projections of Group Algebras....Pages 461-476
Counterexamples for Validation and Discovering of New Theorems....Pages 477-490
The Making of GABLE: A Geometric Algebra Learning Environment in Matlab....Pages 491-511
Helmstetter Formula and Rigid Motions with CLIFFORD....Pages 512-534
Back Matter....Pages 535-592

✦ Subjects

Applications of Mathematics; Mathematics of Computing; Algebra; Appl.Mathematics/Computational Methods of Engineering; Computational Intelligence; Control, Robotics, Mechatronics