Geometric Algebra with Applications in Science and Engineering

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.

This book is addressed to a broad audience of cyberneticists, computer scientists, engineers, applied physicists and applied mathematicians. The book offers several examples to clarify the importance of geometric algebra in signal and image processing, filtering and neural computing, computer vision, robotics and geometric physics. The contributions of this book will help the reader to greater understand the potential of geometric algebra for the design and implementation of real time artifical systems.

I Advances in Geometric Algebra.- 1 Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry.- 2 Universal Geometric Algebra.- 3 Realizations of the Conformal Group.- 4 Hyperbolic Geometry.- II Theorem Proving.- 5 Geometric Reasoning With Geometric Algebra.- 6 Automated Theorem Proving.- III Computer Vision.- 7 The Geometry Algebra of Computer Vision.- 8 Using Geometric Algebra for Optical Motion Capture.- 9 Bayesian Inference and Geometric Algebra: An Application to Camera Localization.- 10 Projective Reconstruction of Shape and Motion Using Invariant Theory.- IV Robotics.- 11 Robot Kinematics and Flags.- 12 The Clifford Algebra and the Optimization of Robot Design.- 13 Applications of Lie Algebras and the Algebra of Incidence.- V Quantum and Neural Computing, and Wavelets.- 14 Geometric Algebra in Quantum Information Processing by Nuclear Magnetic Resonance.- 15 Geometric Feedforward Neural Networks and Support Mul- tivector Machines.- 16 Image Analysis Using Quaternion Wavelets.- VI Applications to Engineering and Physics.- 17 Objects in Contact: Boundary Collisions as Geometric Wave Propagation.- 18 Modern Geometric Calculations in Crystallography.- 19 Quaternion Optimization Problems in Engineering.- 20 Clifford Algebras in Electrical Engineering.- 21 Applications of Geometric Algebra in Physics and Links With Engineering.- VII Computational Methods in Clifford Algebras.- 22 Clifford Algebras as Projections of Group Algebras.- 23 Counterexamples for Validation and Discovering of New Theorems.- 24 The Making of GABLE: A Geometric Algebra Learning Environment in Matlab.- 25 Helmstetter Formula and Rigid Motions with CLIFFORD.- References.