Applications of Geometric Algebra in Computer Science and Engineering

by Leo Dorst (Editor), Chris Doran (Editor), Joan Lasenby (Editor)

Overview

Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant ideas are introduced in a self-contained manner and only a knowledge of linear algebra and calculus is assumed. Features and Topics: * The mathematical foundations of geometric algebra are explored * Applications in computational geometry include models of reflection and ray-tracing and a new and concise characterization of the crystallographic groups * Applications in engineering include robotics, image geometry, control-pose estimation, inverse kinematics and dynamics, control and visual navigation * Applications in physics include rigid-body dynamics, elasticity, and electromagnetism * Chapters dedicated to quantum information theory dealing with multi- particle entanglement, MRI, and relativistic generalizations Practitioners, professionals, and researchers working in computer science, engineering, physics, and mathematics will find a wide range of useful applications in this state-of-the-art survey and reference book. Additionally, advanced graduate students interested in geometric algebra will find the most current applications and methods discussed.

Product Details

ISBN-13:	9781461266068
Publisher:	Birkh�user Boston
Publication date:	10/23/2012
Edition description:	Softcover reprint of the original 1st ed. 2002
Pages:	478
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

1 Point Groups and Space Groups in Geometric Algebra.- 2 The Inner Products of Geometric Algebra.- 3 Unification of Grassmann’s Progressive and Regressive Products using the Principle of Duality.- 4 From Unoriented Subspaces to Blade Operators.- 5 Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra.- 6 Rotations in n Dimensions as Spherical Vectors.- 7 Geometric and Algebraic Canonical Forms.- 8 Functions of Clifford Numbers or Square Matrices.- 9 Compound Matrices and PfafRans: A Representation of Geometric Algebra.- 10 Analysis Using Abstract Vector Variables.- 11 A Multivector Data Structure for Differential Forms and Equations.- 12 Jet Bundles and the Formal Theory of Partial Differential Equations.- 13 Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry.- 14 Symbolic Processing of Clifford Numbers in C++.- 15 Clifford Numbers and their Inverses Calculated using the Matrix Representation.- 16 A Toy Vector Field Based on Geometric Algebra.- 17 Quadratic Transformations in the Projective Plane.- 18 Annihilators of Principal Ideals in the Grassmann Algebra.- 19 Homogeneous Rigid Body Mechanics with Elastic Coupling.- 20 Analysis of One and Two Particle Quantum Systems using Geometric Algebra.- 21 Interaction and Entanglement in the Multiparticle Spacetime Algebra.- 22 Laws of Reflection from Two or More Plane Mirrors in Succession.- 23 Exact Kinetic Energy Operators for Polyatomic Molecules.- 24 Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles.- 25 Is the Brain a ‘Clifford Algebra Quantum Computer’?.- 26 A Hestenes Spacetime Algebra Approach to Light Polarization.- 27 Quaternions, Clifford Algebra and Symmetry Groups.- 28 A Generic Framework for Image Geometry.- 29 Color Edge DetectionUsing Rotors.- 30 Numerical Evaluation of Versors with Clifford Algebra.- 31 The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems.- 32 Applications of Algebra of Incidence in Visually Guided Robotics.- 33 Monocular Pose Estimation of Kinematic Chains.- 34 Stabilization of 3D Pose Estimation.- 35 Inferring Dynamical Information from 3D Position Data using Geometric Algebra.- 36 Clifford Algebra Space Singularities of Inline Planar Platforms.- 37 Fast Quantum Fourier—Heisenberg—Weyl Transforms.- 38 The Structure Multivector.- 39 The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition.- 40 An Algorithm to Solve the Inverse IFS-Problem.- 41 Fast Quantum n-D Fourier and Radon Transforms.

Editorial Reviews

"This book contains papers presented at the conference "Applied Geometric Algebra in Computer Science and Engineering" (AGACSE 2001)…. The goal was to demonstrate how the framework of geometric algebra (Clifford algebra) could unify and illuminate diverse fields of science and engineering.

The articles reveal [a] range [of] fields: from quantum physics to robotics, from crystallographic groups to image understanding, from relativistic mechanics to signal processing. Despite this diversity, the combination of these subjects was not felt to be artificial.

This book should be…useful to mathematicians…physicists, [and] to mechanical and computer engineers."

—Iasi Polytechnic Magazine

"The conference ‘Applied Geometric Algebras in Computer Science and Engineering’ (AGACSE 2001) was held… July 9–13, 2001. The present book contains the papers of this scientific meeting and reflects the constantly growing interest in searching the applications of geometric algebra (or Clifford algebra) in various fields of science.

Geometric algebra includes a lot of techniques from several mathematical theories (linear algebra, vector calculus, projective geometry, complex analysis) and offers new directions in some unexpected domains like quantum physics, robotics, crystallographic groups, image understanding, relativistic mechanics, signal processing.

The volume begins with a preface written by the Editors and a useful list with contributors…. There are four sections: Algebra and Geometry…, Applications to Physics…, Computer Vision and Robotics…, Signal Processing and Other Applications….

In conclusion, a very useful book both for beginners and specialists!"

—Memoriile Sectiilor Stiintifice

From the Publisher