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.
Author(s): Ronald Goldman
Edition: 1
Publisher: CRC Press
Year: 2023
Language: English
Commentary: Publisher PDF
Pages: 278
City: Boca Raton, FL
Tags: Quaternions; Clifford Algebras; Computer Graphics
Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Author
Part I: Dual Quaternions
1.1 Algebras and Dual Algebras
1.2 Algebra
1.2.1 Quaternion Algebra
1.2.2 Conjugates
1.2.3 Dot Products and Norms: Lengths and Angles
1.3 Geometry
1.3.1 Points and Vectors in the Space of Dual Quaternions
1.3.2 Planes in the Space of Dual Quaternions
1.3.3 Lines in the Space of Dual Quaternions
1.3.3.1 Plucker Coordinates
1.3.3.2 Dual Plucker Coordinates
1.3.4 Duality in the Space of Dual Quaternions
1.4 Rigid Motions
1.4.1 Rotation and Translation
1.4.2 Rotations about Arbitrary Lines and Screw Transformations
1.4.3 Screw Transformations and Rigid Motions
1.4.4 Rotation and Translation on Planes
1.4.5 Rotation and Translation on Lines
1.4.6 Reflections
1.5 Rigid Motions as Rotations in 8-Dimensions
1.5.1 Rigid Motions as Linear Isometries in 8-Dimensions
1.5.2 Renormalization
1.6 Screw Linear Interpolation (ScLERP)
1.6.1 Spherical Linear Interpolation (SLERP) Revisited
1.6.2 The Trigonometric Form of the Screw Transformation
1.6.3 ScLERP
1.7 Perspective and Pseudo-Perspective
1.7.1 Perspective in the Quaternion Algebra
1.7.2 Rotation, Translation, and Duality
1.7.3 Perspective Projection
1.7.4 Pseudo-Perspective
1.8 Visualizing Quaternions and Dual Quaternions
1.9 Matrices versus Dual Quaternions
1.9.1 Representations and Computations with Matrices and Dual Quaternions
1.9.2 Converting between Matrices and Dual Quaternions
1.9.2.1 Rigid Motions
1.9.2.2 Perspective and Pseudo-Perspective
1.10 Insights
1.11 Formulas
1.11.1 Algebra
1.11.2 Geometry
1.11.3 Duality
1.11.4 Transformations
1.11.5 Interpolation
1.11.6 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
1 Algebra
2 Geometry
2.1 Planes
2.2 Points and Vectors
2.2.1 Incidence Relation: Point on Plane
2.3 Lines
2.3.1 Lines as the Intersection of Two Planes
2.3.2 Lines as Bivectors
2.3.3 Incidence Relations for Lines
2.4 Duality
2.4.1 Duality in the Quaternion Subalgebra
2.4.2 Duality in the Plane Model
2.4.3 Lines as the Join of Two Points
3 Transformations: Rotors and Versors
3.1 Translation
3.2 Rotation
3.3 Reflection
3.4 Perspective and Pseudo-Perspective
4 Insights
5 Formulas
5.1 Algebra
5.2 Geometry
5.3 Rotors and Versors
5.4 Perspective and Pseudo-Perspective
6 Comparisons between Dual Quaternions and the Plane Model of Clifford Algebra
3 The Point Model of Clifford Algebra for Dual Quaternions
1 Algebra
2 Geometry
2.1 Points and Vectors
2.2 Planes
2.2.1 Incidence Relation: Point on Plane
2.3 Duality
2.3.1 Duality in the Point Model
2.4 Lines
2.4.1 Lines as the Join of Two Points
2.4.2 Lines as Bivectors
2.4.3 Incidence Relations for Lines
2.4.4 Lines as the Intersection of Two Planes
3 Transformations: Rotors and Versors
3.1 Translation
3.2 Rotation
3.3 Reflection
3.4 Perspective and Pseudo-Perspective
4 Insights
5 Formulas
5.1 Algebra
5.2 Geometry
5.3 Rotors and Versors
5.4 Perspective and Pseudo-Perspective
6 Comparisons between the Point Model and the Plane Model of Clifford Algebra
Bibliography
Index
