Synthesis Lectures on Computer Graphics and Animation Rethinking Quaternions - Ron Goldman Kartoniert (TB)

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 areto 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 interpolate between 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