## 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.

2.
3. The Morgan Kaufmann Series in Interactive 3D Technology
5. Foreword
6. Preface
7. Acknowledgments
8. I. Elements of Quaternions
1. 01. The Discovery of Quaternions
2. 02. Folklore of Rotations
3. 03. Basic Notation
4. 04. What Are Quaternions?
5. 05. Road Map to Quaternion Visualization
6. 06. Fundamentals of Rotations
1. 6.1. 2D Rotations
2. 6.2. Quaternions and 3D Rotations
3. 6.3. Recovering θ and
4. 6.4. Euler Angles and Quaternions
5. 6.5. † Optional Remarks
6. 6.6. Conclusion
7. 07. Visualizing Algebraic Structure
1. 7.1. Algebra of Complex Numbers
2. 7.2. Quaternion Algebra
8. 08. Visualizing Spheres
1. 8.1. 2D: Visualizing an Edge-on Circle
2. 8.2. The Square Root Method
3. 8.3. 3D: Visualizing a Balloon
4. 8.4. 4D: Visualizing Quaternion Geometry on S3
9. 09. Visualizing Logarithms and Exponentials
10. 10. Visualizing Interpolation Methods
1. 10.1. Basics of Interpolation
2. 10.2. Quaternion Interpolation
3. 10.3. Equivalent 3×3 Matrix Method
11. 11. Looking at Elementary Quaternion Frames
12. 12. Quaternions and the Belt Trick: Connecting to the Identity
1. 12.1. Very Interesting, but Why?
2. 12.2. The Details: How Quaternion Visualization Explains the Belt Trick
3. 12.3. Frame-sequence Visualization Methods
13. 13. Quaternions and the Rolling Ball: Exploiting Order Dependence
14. 14. Quaternions and Gimbal Lock: Limiting the Available Space
1. 15. Alternative Ways of Writing Quaternions
2. 16. Efficiency and Complexity Issues
1. 16.1. Extracting a Quaternion
2. 16.2. Efficiency of Vector Operations
1. 17.1. Projective Method
2. 17.2. Distance-preserving Flattening Methods
4. 18. More on Logarithms and Exponentials
5. 19. Two-Dimensional Curves
1. 19.1. Orientation Frames for 2D Space Curves
2. 19.2. What Is a Map?
3. 19.3. Tangent and Normal Maps
4. 19.4. Square Root Form
6. 20. Three-Dimensional Curves
1. 20.1. Introduction to 3D Space Curves
2. 20.2. General Curve Framings in 3D
3. 20.3. Tubing
4. 20.4. Classical Frames
5. 20.5. Mapping the Curvature and Torsion
6. 20.6. Theory of Quaternion Frames
7. 20.7. Assigning Smooth Quaternion Frames
8. 20.8. Examples: Torus Knot and Helix Quaternion Frames
9. 20.9. Comparison of Quaternion Frame Curve Lengths
7. 21. 3D Surfaces
1. 21.1. Introduction to 3D Surfaces
2. 21.2. Quaternion Weingarten Equations
3. 21.3. Quaternion Gauss Map
4. 21.4. Example: The Sphere
5. 21.5. Examples: Minimal Surface Quaternion Maps
8. 22. Optimal Quaternion Frames
1. 22.1. Background
2. 22.2. Motivation
3. 22.3. Methodology
4. 22.4. The Space of Frames
5. 22.5. Choosing Paths in Quaternion Space
6. 22.6. Examples
9. 23. Quaternion Volumes
1. 23.1. Three-degree-of-freedom Orientation Domains
2. 23.2. Application to the Shoulder Joint
3. 23.3. Data Acquisition and the Double-covering Problem
4. 23.4. Application Data
10. 24. Quaternion Maps of Streamlines
1. 24.1. Visualization Methods
2. 24.2. 3D Flow Data Visualizations
3. 24.3. Brushing: Clusters and Inverse Clusters
11. 25. Quaternion Interpolation
1. 25.1. Concepts of Euclidean Linear Interpolation
3. 25.3. Direct Interpolation of 3D Rotations
4. 25.4. Quaternion Splines
5. 25.5. Quaternion de Casteljau Splines
6. 25.6. Equivalent Anchor Points
7. 25.7. Angular Velocity Control
8. 25.8. Exponential-map Quaternion Interpolation
9. 25.9. Global Minimal Acceleration Method
12. 26. Quaternion Rotator Dynamics
13. 27. Concepts of the Rotation Group
1. 27.1. Brief Introduction to Group Representations
2. 27.2. Basic Properties of Spherical Harmonics
14. 28. Spherical Riemannian Geometry
1. 28.1. Induced Metric on the Sphere
2. 28.2. Induced Metrics of Spheres
3. 28.3. Elements of Riemannian Geometry
4. 28.4. Riemann Curvature of Spheres
5. 28.5. Geodesics and Parallel Transport on the Sphere
6. 28.6. Embedded-vector Viewpoint of the Geodesics
10. III. Beyond Quaternions
1. 29. The Relationship of 4D Rotations to Quaternions
2. 30. Quaternions and the Four Division Algebras
1. 30.1. Division Algebras
2. 30.2. Relation to Fiber Bundles
3. 30.3. Constructing the Hopf Fibrations
3. 31. Clifford Algebras
1. 31.1. Introduction to Clifford Algebras
2. 31.2. Foundations
3. 31.3. Examples of Clifford Algebras
4. 31.4. Higher Dimensions
5. 31.5. Pin(N), Spin(N), O(N), SO(N), and All That...
4. 32. Conclusions
11. Appendices
1. A. Notation
1. A.1. Vectors
2. A.2. Length of a Vector
3. A.3. Unit Vectors
4. A.4. Polar Coordinates
5. A.5. Spheres
6. A.6. Matrix Transformations
7. A.7. Features of Square Matrices
8. A.8. Orthogonal Matrices
9. A.9. Vector Products
10. A.10. Complex Variables
2. B. 2D Complex Frames
3. C. 3D Quaternion Frames
4. D. Frame and Surface Evolution
5. E. Quaternion Survival Kit
6. F. Quaternion Methods
1. F.1. Quaternion Logarithms and Exponentials
2. F.2. The Quaternion Square Root Trick
3. F.3. The â → Formula Simplified
4. F.4. Gram–Schmidt Spherical Interpolation
5. F.5. Direct Solution for Spherical Interpolation
6. F.6. Converting Linear Algebra to Quaternion Algebra
7. F.7. Useful Tensor Methods and Identities
7. G. Quaternion Path Optimization Using Surface Evolver
8. H. Quaternion Frame Integration
9. I. Hyperspherical Geometry
12. References

## Product Information

• Title: The Morgan Kaufmann Series in Interactive 3D Technology: Visualizing Quaternions
• Author(s): Andrew J. Hanson
• Release date: January 2006
• Publisher(s): Morgan Kaufmann
• ISBN: 9780120884001