Clifford algebra for dual quaternions has emerged recently as an alternative to standard matrix algebra as a computational framework for computer graphics. This book presents dual quaternions and their associated Clifford algebras in a new light, accessible to and geared toward the computer graphics community.
Collecting all the associated formulas and theorems in one place, this book provides an extensive and rigorous treatment of dual quaternions, as well as showing how two models of Clifford algebra emerge naturally from the theory of dual quaternions. Each section comes complete with a set of exercises to help readers sharpen and practice their understanding.
This book is accessible to anyone with a basic knowledge of quaternion algebra and is of particular use to forward-thinking members of the computer graphics community.
By:
Ronald Goldman (Rice University Houston Texas USA)
Imprint: CRC Press
Country of Publication: United Kingdom
Dimensions:
Height: 229mm,
Width: 152mm,
ISBN: 9781032502977
ISBN 10: 1032502975
Pages: 258
Publication Date: 19 December 2024
Audience:
College/higher education
,
Primary
Format: Paperback
Publisher's Status: Forthcoming
Contents Preface Part I: Dual Quaternions Algebras and Dual Algebras 2. Algebra 2.1 Quaternion Algebra 2.2 Conjugates 2.3 Dot Products and Norms: Lengths and Angles 3. Geometry 3.1 Points and Vectors in the Space of Dual Quaternions 3.2 Planes in the Space of Dual Quaternions 3.3 Lines in the Space of Dual Quaternions 3.3.1 Plucker Coordinates 3.3.2 Dual Plucker Coordinates 3.4 Duality in the Space of Dual Quaternions 4. Rigid Motions 4.1 Rotation and Translation 4.2 Rotations about Arbitrary Lines and Screw Transformations 4.3 Screw Transformations and Rigid Motions 4.4 Rotation and Translation on Planes 4.5 Rotation and Translation on Lines 4.6 Reflections 5. Rigid Motions as Rotations in 8-Dimensions 5.1 Rigid Motions as Linear Isometries in 8-Dimensions 5.2 Renormalization 6. Screw Linear Interpolation (ScLERP) 6.1 Spherical Linear Interpolation (SLERP) Revisited 6.2 The Trigonometric Form of the Screw Transformation 6.3 ScLERP 7. Perspective and Pseudo-Perspective 7.1 Perspective in the Quaternion Algebra 7.2 Rotation, Translation, and Duality 7.3 Perspective Projection 7.4 Pseudo-Perspective 8. Visualizing Quaternions and Dual Quaternions 9. Matrices vs. Dual Quaternions 9.1 Representations and Computations with Matrices and Dual Quaternions 9.2 Converting between Matrices and Dual Quaternions 9.2.1 Rigid Motions 9.2.2 Perspective and Pseudo-Perspective 10. Insights 11. Formulas 11.1 Algebra 11.2 Geometry 11.3 Duality 11.4 Transformations 11.5 Interpolation 11.6 Conversion Formulas Appendix: Cross Products Part II: Clifford Algebras for Dual Quaternions 1: A Brief Review of Clifford Algebra 1. Goals of Clifford Algebra 2. A Brief Introduction to Clifford Algebra 3. Basic Products: Clifford Product, Inner Product, and Outer Product 3.1 Exterior Algebra: The Outer (Wedge) Product for Arbitrary Grades 4. Duality 4.1 Duality in the Quaternion Algebra: Cross Products and Products of Pure Quaternions 2: The Plane Model of Clifford Algebra for Dual Quaternions 2.1. Algebra 2.2. Geometry 2.2.1 Planes 2.2.2 Points and Vectors 2.2.2.1 Incidence Relation: Point on Plane 2.2.3 Lines 2.2.3.1 Lines as the Intersection of Two Planes 2.2.3.2 Lines as Bivectors 2.2.3.3 Incidence Relations for Lines 2.2.4 Duality 2.2.4.1 Duality in the Quaternion Subalgebra 2.2.4.2 Duality in the Plane Model 2.2.4.3 Lines as the Join of Two Points 2.3. Transformations: Rotors and Versors 2.3.1 Translation 2.3.2 Rotation 2.3.3 Reflection 2.3.4 Perspective and Pseudo-Perspective 2.4. Insights 2.5. Formulas 2.5.1 Algebra 2.5.2 Geometry 2.5.3 Rotors and Versors 2.5.4 Perspective and Pseudo-Perspective 2.6. Comparisons Between Dual Quaternions and the Plane Model of Clifford Algebra 3. The Point Model of Clifford Algebra for Dual Quaternions 3.1. Algebra 3.2. Geometry 3.2.1 Points and Vectors 3.2.2 Planes 3.2.2.1 Incidence Relation: Point on Plane 3.2.3 Duality 3.2.3.1 Duality in the Point Model 3.2.4 Lines 3.2.4.1 Lines as the Join of Two Points 3.2.4.2 Lines as Bivectors 3.2.4.3 Incidence Relations for Lines 3.2.4.4 Lines as the Intersection of Two Planes 3.3. Transformations: Rotors and Versors 3.3.1 Translation 3.3.2 Rotation 3.3.3 Reflection 3.3.4 Perspective and Pseudo-Perspective 3.4. Insights 3.5. Formulas 3.5.1 Algebra 3.5.2 Geometry 3.5.3 Rotors and Versors 3.5.4 Perspective and Pseudo-Perspective 3.6. Comparisons Between the Point Model and the Plane Model of Clifford Algebra Bibliography
Ronald Goldman is a Professor of Computer Science at Rice University in Houston, Texas. Professor Goldman received his B.S. in Mathematics from the Massachusetts Institute of Technology in 1968 and his M.A. and Ph.D. in Mathematics from Johns Hopkins University in 1973. Professor Goldman’s current research concentrates on the mathematical representation, manipulation, and analysis of shape using computers. His work includes research in computer-aided geometric design, solid modeling, computer graphics, subdivision, polynomials, and splines. His most recent focus is on the uses of quaternions, dual quaternions, and Clifford algebras in computer graphics. Dr. Goldman has published over two hundred research articles in journals, books, and conference proceedings. He has also authored two books on computer graphics and geometric modeling: Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and An Integrated Introduction to Computer Graphics and Geometric Modeling. In addition, he has written an extended monograph on quaternions: Rethinking Quaternions: Theory and Computation. Dr. Goldman is currently an Associate Editor of Computer Aided Geometric Design. Before returning to academia, Dr. Goldman worked for ten years in industry solving problems in computer graphics, geometric modeling, and computer-aided design. He served as a Mathematician at Manufacturing Data Systems Inc., where he helped to implement one of the first industrial solid modeling systems. Later he worked as a Senior Design Engineer at Ford Motor Company, enhancing the capabilities of their corporate graphics and computer-aided design software. From Ford he moved on to Control Data Corporation, where he was a Principal Consultant for the development group devoted to computer-aided design and manufacture. His responsibilities included database design, algorithms, education, acquisitions, and research. Dr. Goldman left Control Data Corporation in 1987 to become an Associate Professor of Computer Science at the University of Waterloo in Ontario, Canada. He joined the faculty at Rice University in Houston,Texas as a Professor of Computer Science in July 1990.
