Applications of Geometric Algebra in Computer Science and Engineering

1. Front Matter

   Pages i-xxv

   PDF

2. Algebra and Geometry

   1. Front Matter

      Pages 1-1

      PDF

   2. Point Groups and Space Groups in Geometric Algebra

      • David Hestenes

      Pages 3-34

   3. The Inner Products of Geometric Algebra

      • Leo Dorst

      Pages 35-46

   4. Unification of Grassmann’s Progressive and Regressive Products using the Principle of Duality

      • Stephen Blake

      Pages 47-57

   5. From Unoriented Subspaces to Blade Operators

      • Timaeus A. Bouma

      Pages 59-67

   6. Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra

      • Hongbo Li

      Pages 69-78

   7. Rotations in n Dimensions as Spherical Vectors

      • W. E. Baylis, S. Hadi

      Pages 79-90

   8. Geometric and Algebraic Canonical Forms

      • Neil Gordon

      Pages 91-98

   9. Functions of Clifford Numbers or Square Matrices

      • John Snygg

      Pages 99-107

   10. Compound Matrices and Pfaffians: A Representation of Geometric Algebra

       • Uwe Prells, Michael I. Friswell, Seamus D. Garvey

       Pages 109-118

   11. Analysis Using Abstract Vector Variables

       • Frank Sommen

       Pages 119-128

   12. A Multivector Data Structure for Differential Forms and Equations

       • Jeffrey A. Chard, Vadim Shapiro

       Pages 129-131

   13. Jet Bundles and the Formal Theory of Partial Differential Equations

       • Richard Baker, Chris J. L. Doran

       Pages 133-143

   14. Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry

       • Eckhard M. S. Hitzer

       Pages 145-155

   15. Symbolic Processing of Clifford Numbers in C++

       • John P. Fletcher

       Pages 157-167

   16. Clifford Numbers and their Inverses Calculated using the Matrix Representation

       • John P. Fletcher

       Pages 169-178

   17. A Toy Vector Field Based on Geometric Algebra

       • Alyn Rockwood, Shoeb Binderwala

       Pages 179-185

   18. Quadratic Transformations in the Projective Plane

       • Georgi Georgiev

       Pages 187-191

   19. Annihilators of Principal Ideals in the Grassmann Algebra

       • Cemal Koc, Songul Esin

       Pages 193-194

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.

• Algebra
• algorithm
• calculus
• differential equation
• electromagnetism
• geometry
• information theory
• kinematics
• linear algebra
• mechanics
• model
• operator
• partial differential equation
• robotics
• transformation

"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

• Informatics Institute, University of Amsterdam, Amsterdam, The Netherlands

  Leo Dorst

• Cavendish Laboratory, Cambridge University, Cambridge, UK

  Chris Doran

• Department of Engineering-Signal Processing, Cambridge University, Cambridge, UK

  Joan Lasenby

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