Applications of Geometric Algebra in Computer Science and Engineering / Edition 1

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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.
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Editorial Reviews

From the Publisher

"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

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Product Details

  • ISBN-13: 9780817642679
  • Publisher: Birkhauser Verlag
  • Publication date: 3/1/2002
  • Edition description: 2002
  • Edition number: 1
  • Pages: 478
  • Product dimensions: 6.26 (w) x 9.48 (h) x 1.02 (d)

Table of Contents

Preface
Contributors
I Algebra and Geometry 1
1 Point Groups and Space Groups in Geometric Algebra 3
2 The Inner Products of Geometric Algebra 35
3 Unification of Grassmann's Progressive and Regressive Products using the Principle of Duality 47
4 From Unoriented Subspaces to Blade Operators 59
5 Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra 69
6 Rotations in n Dimensions as Spherical Vectors 79
7 Geometric and Algebraic Canonical Forms 91
8 Functions of Clifford Numbers of Square Matrices 99
9 Compound Matrices and Pfaffians: A Representation of Geometric Algebra 109
10 Analysis Using Abstract Vector Variables 119
11 A Multivector Data Structure for Differential Forms and Equations 129
12 Jet Bundles and the Formal Theory of Partial Differential Equations 133
13 Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry 145
14 Symbolic Processing of Clifford Numbers in C++ 157
15 Clifford Numbers and their Inverses Calculated using the Matrix Representation 169
16 A Toy Vector Field Based on Geometric Algebra 179
17 Quadratic Transformations in the Projective Plane 187
18 Annihilators of Principal Ideals in the Grassmann Algebra 193
II Applications to Physics 195
19 Homogeneous Rigid Body Mechanics with Elastic Coupling 197
20 Analysis of One and Two Particle Quantum Systems using Geometric Algebra 213
21 Interaction and Entanglement in the Multiparticle Spacetime Algebra 227
22 Laws of Reflection from Two or More Plane Mirrors in Succession 249
23 Exact Kinetic Energy Operators for Polyatomic Molecules 261
24 Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles 271
25 Is the Brain a 'Clifford Algebra Quantum Computer'? 285
26 A Hestenes Spacetime Algebra Approach to Light Polarization 297
27 Quaternions, Clifford Algebra and Symmetry Groups 307
III Computer Vision and Robotics 317
28 A Generic Framework for Image Geometry 319
29 Color Edge Detection Using Rotors 333
30 Numerical Evaluation of Versors with Clifford Algebra 341
31 The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems 351
32 Applications of Algebra of Incidence in Visually Guided Robotics 361
33 Monocular Pose Estimation of Kinematic Chains 373
34 Stabilization of 3D Pose Estimation 385
35 Inferring Dynamical Information from 3D Position Data using Geometric Algebra 395
36 Clifford Algebra Space Singularities of Inline Planar Platforms 407
IV Signal Processing and Other Applications 423
37 Fast Quantum Fourier-Heisenberg-Weyl Transforms 425
38 The Structure Multivector 437
39 The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition 449
40 An Algorithm to Solve the Inverse IFS-Problem 459
41 Fast Quantum n-D Fourier and Radon Transforms 469
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