Book description
Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available.The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important—a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the allimportant advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their fourdimensional nature and to Clifford Algebras, the allencompassing framework for vectors and quaternions.
* Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing.
* Covers both nonmathematical and mathematical approaches to quaternions.
* Companion website with an assortment of quaternion utilities and sample code, data sets for the book's illustrations, and Mathematica notebooks with essential algebraic utilities.
Table of contents
 Front Cover (1/2)
 Front Cover (2/2)
 Visualizing Quaternions
 Copyright Page
 Contents
 About the Author
 Foreword
 Preface (1/2)
 Preface (2/2)
 Acknowledgments

Part I: Elements of Quaternions
 Chapter 1. The Discovery of Quaternions
 Chapter 2. Folklore of Rotations
 Chapter 3. Basic Notation
 Chapter 4. What are Quaternions?
 Chapter 5. Road Map to Quaternion Visualization
 Chapter 6. Fundamentals of Rotations
 Chapter 7. Visualizing Algebraic Structure
 Chapter 8. Visualizing Spheres
 Chapter 9. Visualizing Logarithms and Exponentials
 Chapter 10. Visualizing Interpolation Methods
 Chapter 11. Looking at Elementary Quaternion Frames
 Chapter 12. Quaternions and the Belt Trick: Connecting to the Identity
 Chapter 13. Quaternions and the Rolling Ball: Exploiting Order Dependence
 Chapter 14. Quaternions and Gimbal Lock: Limiting the Available Space

Part II: Advanced Quaternion Topics
 Chapter 15. Alternative Ways of Writing Quaternions
 Chapter 16. Efficiency and Complexity Issues
 Chapter 17. Advanced Sphere Visualization
 Chapter 18. More on Logarithms and Exponentials
 Chapter 19. TwoDimensional Curves

Chapter 20. ThreeDimensional Curves
 20.1 Introduction to 3D Space Curves
 20.2 General Curve Framings in 3D
 20.3 Tubing
 20.4 Classical Frames
 20.5 Mapping the Curvature and Torsion
 20.6 Theory of Quaternion Frames (1/2)
 20.6 Theory of Quaternion Frames (2/2)
 20.7 Assigning Smooth Quaternion Frames (1/2)
 20.7 Assigning Smooth Quaternion Frames (2/2)
 20.8 Examples: Torus Knot and Helix Quaternion Frames
 20.9 Comparison of Quaternion Frame Curve Lengths
 Chapter 21. 3D Surfaces
 Chapter 22. Optimal Quaternion Frames
 Chapter 23. Quaternion Volumes
 Chapter 24. Quaternion Maps of Streamlines

Chapter 25. Quaternion Interpolation
 25.1 Concepts of Euclidean Linear Interpolation (1/2)
 25.1 Concepts of Euclidean Linear Interpolation (2/2)
 25.2 The Double Quad
 25.3 Direct Interpolation of 3D Rotations (1/2)
 25.3 Direct Interpolation of 3D Rotations (2/2)
 25.4 Quaternion Splines
 25.5 Quaternion de Casteljau Splines (1/2)
 25.5 Quaternion de Casteljau Splines (2/2)
 25.6 Equivalent Anchor Points
 25.7 Angular Velocity Control
 25.8 Exponentialmap Quaternion Interpolation
 25.9 Global Minimal Acceleration Method
 Chapter 26. Quaternion Rotator Dynamics
 Chapter 27. Concepts of the Rotation Group
 Chapter 28. Spherical Riemannian Geometry
 Part III: Beyond Quaternions
 Appendices
 References (1/4)
 References (2/4)
 References (3/4)
 References (4/4)
 Index (1/3)
 Index (2/3)
 Index (3/3)
Product information
 Title: Visualizing Quaternions
 Author(s):
 Release date: February 2006
 Publisher(s): Elsevier Science
 ISBN: None
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