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 all-important 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 four-dimensional nature and to
Clifford Algebras, the all-encompassing framework for vectors and
quaternions.
* Richly illustrated introduction for the developer, scientist,
engineer, or student in computer graphics, visualization, or
entertainment computing.
* Covers both non-mathematical 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
- Copyright
- The Morgan Kaufmann Series in Interactive 3D Technology
- About the Author
- Foreword
- Preface
- Acknowledgments
-
I. Elements of Quaternions
- 01. The Discovery of Quaternions
- 02. Folklore of Rotations
- 03. Basic Notation
- 04. What Are Quaternions?
- 05. Road Map to Quaternion Visualization
- 06. Fundamentals of Rotations
- 07. Visualizing Algebraic Structure
- 08. Visualizing Spheres
- 09. Visualizing Logarithms and Exponentials
- 10. Visualizing Interpolation Methods
- 11. Looking at Elementary Quaternion Frames
- 12. Quaternions and the Belt Trick: Connecting to the Identity
- 13. Quaternions and the Rolling Ball: Exploiting Order Dependence
- 14. Quaternions and Gimbal Lock: Limiting the Available Space
-
II. Advanced Quaternion Topics
- 15. Alternative Ways of Writing Quaternions
- 16. Efficiency and Complexity Issues
- 17. Advanced Sphere Visualization
- 18. More on Logarithms and Exponentials
- 19. Two-Dimensional Curves
-
20. Three-Dimensional 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
- 20.7. Assigning Smooth Quaternion Frames
- 20.8. Examples: Torus Knot and Helix Quaternion Frames
- 20.9. Comparison of Quaternion Frame Curve Lengths
- 21. 3D Surfaces
- 22. Optimal Quaternion Frames
- 23. Quaternion Volumes
- 24. Quaternion Maps of Streamlines
-
25. Quaternion Interpolation
- 25.1. Concepts of Euclidean Linear Interpolation
- 25.2. The Double Quad
- 25.3. Direct Interpolation of 3D Rotations
- 25.4. Quaternion Splines
- 25.5. Quaternion de Casteljau Splines
- 25.6. Equivalent Anchor Points
- 25.7. Angular Velocity Control
- 25.8. Exponential-map Quaternion Interpolation
- 25.9. Global Minimal Acceleration Method
- 26. Quaternion Rotator Dynamics
- 27. Concepts of the Rotation Group
- 28. Spherical Riemannian Geometry
-
III. Beyond Quaternions
- 29. The Relationship of 4D Rotations to Quaternions
- 30. Quaternions and the Four Division Algebras
- 31. Clifford Algebras
- 32. Conclusions
-
Appendices
- A. Notation
- B. 2D Complex Frames
- C. 3D Quaternion Frames
- D. Frame and Surface Evolution
- E. Quaternion Survival Kit
- F. Quaternion Methods
- G. Quaternion Path Optimization Using Surface Evolver
- H. Quaternion Frame Integration
- I. Hyperspherical Geometry
- References
Product information
- Title: The Morgan Kaufmann Series in Interactive 3D Technology: Visualizing Quaternions
- Author(s):
- Release date: January 2006
- Publisher(s): Morgan Kaufmann
- ISBN: 9780120884001
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