Dual Quaternions and Their Associated Clifford Algebras

图书简介

Amid recent interest in Clifford algebra for dual quaternions as a more suitable method for Computer Graphics than standard matrix algebra, this book presents dual quaternions and their associated Clifford algebras in a new light, accessible to and geared towards the Computer Graphics community.Collating 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 algebras emerge naturally from the theory of dual quaternions. Each chapter comes complete with a set of exercises to help readers sharpen and practice their knowledge. 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. .

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 1

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Dual Quaternions and Their Associated Clifford Algebras 对偶四元数及其相关克利福德代数 计算机科学技术基础学科

Dual Quaternions and Their Associated Clifford Algebras

对偶四元数及其相关克利福德代数
计算机科学技术基础学科