Rethinking Quaternions

Quaternion multiplication can be used to rotate vectors in three-dimensions. Therefore, in computer graphics, quaternions have three principal applications: to increase speed and reduce storage for calculations involving rotations, to avoid distortions arising from numerical inaccuracies caused by floating point computations with rotations, and to interpolate between two rotations for key frame animation. Yet while the formal algebra of quaternions is well-known in the graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this monograph are to provide a fresh, geometric interpretation for quaternions, appropriate for contemporary computer graphics, based on mass-points; to present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in three dimensions using insights from the algebra and geometry of multiplication in the complex plane; to derive the formula for quaternion multiplication from first principles; to develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection; to show how to apply sandwiching to compute perspective projections. In addition to these theoretical issues, we also address some computational questions. We develop straightforward formulas for converting back and forth between quaternion and matrix representations for rotations, reflections, and perspective projections, and we discuss the relative advantages and disadvantages of the quaternion and matrix representations for these transformations. Moreover, we show how to avoid distortions due to floating point computations with rotations by using unit quaternions to represent rotations. We also derive the formula for spherical linear interpolation, and we explain how to apply this formula to interpolatebetween two rotations for key frame animation. Finally, we explain the role of quaternions in low-dimensional Clifford algebras, and we show how to apply the Clifford algebra for R3 to model rotations, reflections, and perspective projections. To help the reader understand the concepts and formulas presented here, we have incorporated many exercises in order to clarify and elaborate some of the key points in the text. Table of Contents: Preface / Theory / Computation / Rethinking Quaternions and Clif ford Algebras / References / Further Reading / Author Biography

Ron Goldman is a professor of computer science at Rice University in Houston, Texas. Dr. 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. Dr. Goldman’s current research interests lie in the mathematical representation, manipulation, and analysis of shape using computers. His work includes research in computer-aided geometric design, solid modeling, computer graphics, polynomials, and splines. He is particularly interested in algorithms for polynomial and piecewise polynomial curves and surfaces, as well as in applications of algebraic and differential geometry to geometric modeling. His most recent focus is on the uses of quaternions and Clifford algebras in computer graphics. Dr. Goldman has published over a hundred articles in journals, books, and conference proceedings on these and related topics. He has also published 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. Dr. Goldman is currently an associate editor of Computer Aided Geometric Design. Before returning to academia, Dr. Goldman worked for 10 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.