Dual Quaternions and Their Associated Clifford Algebras  book cover
1st Edition

Dual Quaternions and Their Associated Clifford Algebras




  • Available for pre-order on June 15, 2023. Item will ship after July 6, 2023
ISBN 9781032502960
July 6, 2023 Forthcoming by CRC Press
146 Pages 1 Color & 21 B/W Illustrations

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Book Description

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.

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Table of Contents

Contents

Preface

Part I: Dual Quaternions

  1. 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

11.5 Interpolation

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

2.1. Algebra

2.2. Geometry

2.2.1 Planes

2.2.2 Points and Vectors

2.2.2.1 Incidence Relation: Point on Plane

2.2.3 Lines

2.2.3.1 Lines as the Intersection of Two Planes

2.2.3.2 Lines as Bivectors

2.2.3.3 Incidence Relations for Lines

2.2.4 Duality

2.2.4.1 Duality in the Quaternion Subalgebra

2.2.4.2 Duality in the Plane Model

2.2.4.3 Lines as the Join of Two Points

2.3. Transformations: Rotors and Versors

2.3.1 Translation

2.3.2 Rotation

2.3.3 Reflection

2.3.4 Perspective and Pseudo-Perspective

2.4. Insights

2.5. Formulas

2.5.1 Algebra

2.5.2 Geometry

2.5.3 Rotors and Versors

2.5.4 Perspective and Pseudo-Perspective

2.6. Comparisons Between Dual Quaternions and the Plane Model of Clifford Algebra

3. The Point Model of Clifford Algebra for Dual Quaternions

3.1. Algebra

3.2. Geometry

3.2.1 Points and Vectors

3.2.2 Planes

3.2.2.1 Incidence Relation: Point on Plane

3.2.3 Duality

3.2.3.1 Duality in the Point Model

3.2.4 Lines

3.2.4.1 Lines as the Join of Two Points

3.2.4.2 Lines as Bivectors

3.2.4.3 Incidence Relations for Lines

3.2.4.4 Lines as the Intersection of Two Planes

3.3. Transformations: Rotors and Versors

3.3.1 Translation

3.3.2 Rotation

3.3.3 Reflection

3.3.4 Perspective and Pseudo-Perspective

3.4. Insights

3.5. Formulas

3.5.1 Algebra

3.5.2 Geometry

3.5.3 Rotors and Versors

3.5.4 Perspective and Pseudo-Perspective

3.6. Comparisons Between the Point Model and the Plane Model of Clifford Algebra

Bibliography

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