Dual Quaternions and Their Associated Clifford Algebras

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.

Part I. Dual Quaternions, 1.1. Algebras and Dual Algebras, 1.2. Algebra, 1.3. Geometry, 1.4. Rigid Motions, 1.5. Rigid Motions as Rotations in 8-Dimensions, 1.6. Screw Linear Interpolation (ScLERP), 1.7. Perspective and Pseudo-Perspective, 1.8. Visualizing Quaternions and Dual Quaternions, 1.9. Matrices versus Dual Quaternions, 1.10. Insights, 1.11. Formulas, Part II. Clifford Algebras for Dual Quaternions, 1. A Brief Review of Clifford Algebra, 2. The Plane Model of Clifford Algebra for Dual Quaternions, 3. The Point Model of Clifford Algebra for Dual Quaternions