Visualizing Quaternions

Visualizing Quaternions

by Andrew J. Hanson
Released February 2006
Publisher(s): Elsevier Science
ISBN: None

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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

  1. Front Cover (1/2)
  2. Front Cover (2/2)
  3. Visualizing Quaternions
  4. Copyright Page
  5. Contents
  6. About the Author
  7. Foreword
  8. Preface (1/2)
  9. Preface (2/2)
  10. Acknowledgments
  11. Part I: Elements of Quaternions
    1. Chapter 1. The Discovery of Quaternions
      1. 1.1 Hamilton's Walk
      2. 1.2 Then Came Octonions
      3. 1.3 The Quaternion Revival
    2. Chapter 2. Folklore of Rotations
      1. 2.1 The Belt Trick
      2. 2.2 The Rolling Ball
      3. 2.3 The Apollo 10 Gimbal-lock Incident (1/2)
      4. 2.3 The Apollo 10 Gimbal-lock Incident (2/2)
      5. 2.4 3D Game Developer's Nightmare
      6. 2.5 The Urban Legend of the Upside-down F16
      7. 2.6 Quaternions to the Rescue
    3. Chapter 3. Basic Notation
      1. 3.1 Vectors
      2. 3.2 Length of a Vector
      3. 3.3 3D Dot Product
      4. 3.4 3D Cross Product
      5. 3.5 Unit Vectors
      6. 3.6 Spheres
      7. 3.7 Matrices
      8. 3.8 Complex Numbers
    4. Chapter 4. What are Quaternions?
    5. Chapter 5. Road Map to Quaternion Visualization
      1. 5.1 The Complex Number Connection
      2. 5.2 The Cornerstones of Quaternion Visualization
    6. Chapter 6. Fundamentals of Rotations
      1. 6.1 2D Rotations
      2. 6.2 Quaternions and 3D Rotations
      3. 6.3 Recovering θ and n
      4. 6.4 Euler Angles and Quaternions
      5. 6.5 † Optional Remarks
      6. 6.6 Conclusion
    7. Chapter 7. Visualizing Algebraic Structure
      1. 7.1 Algebra of Complex Numbers
      2. 7.2 Quaternion Algebra (1/2)
      3. 7.2 Quaternion Algebra (2/2)
    8. Chapter 8. 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 (1/2)
      5. 8.4 4D: Visualizing Quaternion Geometry on S3 (2/2)
    9. Chapter 9. Visualizing Logarithms and Exponentials
      1. 9.1 Complex Numbers
      2. 9.2 Quaternions
    10. Chapter 10. Visualizing Interpolation Methods
      1. 10.1 Basics of Interpolation
      2. 10.2 Quaternion Interpolation
      3. 10.3 Equivalent 3 x 3 Matrix Method
    11. Chapter 11. Looking at Elementary Quaternion Frames
      1. 11.1 A Single Quaternion Frame
      2. 11.2 Several Isolated Frames
      3. 11.3 A Rotating Frame Sequence
      4. 11.4 Synopsis
    12. Chapter 12. Quaternions and the Belt Trick: Connecting to the Identity
      1. 12.1 Very Interesting, but Why?
      2. 12.2 The Details
      3. 12.3 Frame-sequence Visualization Methods
    13. Chapter 13. Quaternions and the Rolling Ball: Exploiting Order Dependence
      1. 13.1 Order Dependence
      2. 13.2 The Rolling Ball Controller
      3. 13.3 Rolling Ball Quaternions
      4. 13.4 Commutators
      5. 13.5 Three degrees of freedom from two
    14. Chapter 14. Quaternions and Gimbal Lock: Limiting the Available Space
      1. 14.1 Guidance System Suspension
      2. 14.2 Mathematical Interpolation Singularities
      3. 14.3 Quaternion Viewpoint
  12. Part II: Advanced Quaternion Topics
    1. Chapter 15. Alternative Ways of Writing Quaternions
      1. 15.1 Hamilton's Generalization of Complex Numbers
      2. 15.2 Pauli Matrices
      3. 15.3 Other Matrix Forms
    2. Chapter 16. Efficiency and Complexity Issues
      1. 16.1 Extracting a Quaternion
      2. 16.2 Efficiency of Vector Operations
    3. Chapter 17. Advanced Sphere Visualization
      1. 17.1 Projective Method
      2. 17.2 Distance-preserving Flattening Methods (1/2)
      3. 17.2 Distance-preserving Flattening Methods (2/2)
    4. Chapter 18. More on Logarithms and Exponentials
      1. 18.1 2D Rotations
      2. 18.2 3D Rotations
      3. 18.3 Using Logarithms for Quaternion Calculus
      4. 18.4 Quaternion Interpolations Versus Log
    5. Chapter 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. Chapter 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 (1/2)
      7. 20.6 Theory of Quaternion Frames (2/2)
      8. 20.7 Assigning Smooth Quaternion Frames (1/2)
      9. 20.7 Assigning Smooth Quaternion Frames (2/2)
      10. 20.8 Examples: Torus Knot and Helix Quaternion Frames
      11. 20.9 Comparison of Quaternion Frame Curve Lengths
    7. Chapter 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. Chapter 22. Optimal Quaternion Frames
      1. 22.1 Background
      2. 22.2 Motivation
      3. 22.3 Methodology
      4. 22.4 The Space of Frames (1/2)
      5. 22.4 The Space of Frames (2/2)
      6. 22.5 Choosing Paths in Quaternion Space
      7. 22.6 Examples (1/2)
      8. 22.6 Examples (2/2)
    9. Chapter 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. Chapter 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
      4. 24.4 Advanced Visualization Approaches
    11. Chapter 25. Quaternion Interpolation
      1. 25.1 Concepts of Euclidean Linear Interpolation (1/2)
      2. 25.1 Concepts of Euclidean Linear Interpolation (2/2)
      3. 25.2 The Double Quad
      4. 25.3 Direct Interpolation of 3D Rotations (1/2)
      5. 25.3 Direct Interpolation of 3D Rotations (2/2)
      6. 25.4 Quaternion Splines
      7. 25.5 Quaternion de Casteljau Splines (1/2)
      8. 25.5 Quaternion de Casteljau Splines (2/2)
      9. 25.6 Equivalent Anchor Points
      10. 25.7 Angular Velocity Control
      11. 25.8 Exponential-map Quaternion Interpolation
      12. 25.9 Global Minimal Acceleration Method
    12. Chapter 26. Quaternion Rotator Dynamics
      1. 26.1 Static Frame
      2. 26.2 Torque
      3. 26.3 Quaternion Angular Momentum
    13. Chapter 27. Concepts of the Rotation Group
      1. 27.1 Brief Introduction to Group Representations
      2. 27.2 Basic Properties of Spherical Harmonics (1/2)
      3. 27.2 Basic Properties of Spherical Harmonics (2/2)
    14. Chapter 28. Spherical Riemannian Geometry
      1. 28.1 Induced Metric on the Sphere
      2. 28.2 Induced Metrics of Spheres (1/2)
      3. 28.2 Induced Metrics of Spheres (2/2)
      4. 28.3 Elements of Riemannian Geometry
      5. 28.4 Riemann Curvature of Spheres
      6. 28.5 Geodesics and Parallel Transport on the Sphere
      7. 28.6 Embedded-vector Viewpoint of the Geodesics
  13. Part III: Beyond Quaternions
    1. Chapter 29. The Relationship of 4D Rotations to Quaternions
      1. 29.1 What Happened in Three Dimensions
      2. 29.2 Quaternions and Four Dimensions
    2. Chapter 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 (1/2)
      4. 30.3 Constructing the Hopf Fibrations (2/2)
    3. Chapter 31. Clifford Algebras
      1. 31.1 Introduction to Clifford Algebras
      2. 31.2 Foundations (1/2)
      3. 31.2 Foundations (2/2)
      4. 31.3 Examples of Clifford Algebras (1/2)
      5. 31.3 Examples of Clifford Algebras (2/2)
      6. 31.4 Higher Dimensions
      7. 31.5 Pin(N), Spin(N), O(N), SO(N), and all that
    4. Chapter 32. Conclusions
  14. 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
      1. C.1 Unit Norm
      2. C.2 Multiplication Rule
      3. C.3 Mapping to 3D rotations
      4. C.4 Rotation Correspondence
      5. C.5 Quaternion Exponential Form
    4. D. Frame and Surface Evolution
      1. D.1 Quaternion Frame Evolution
      2. D.2 Quaternion Surface Evolution
    5. E. Quaternion Survival Kit (1/2)
    6. E. Quaternion Survival Kit (2/2)
    7. F. Quaternion Methods
      1. F.1 Quaternion Logarithms and Exponentials
      2. F.2 The Quaternion Square Root Trick
      3. F.3 The â→b 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
    8. G. Quaternion Path Optimization Using Surface Evolver
    9. H. Quaternion Frame Integration
    10. I. Hyperspherical Geometry
      1. I.1 Definitions
      2. I.2 Metric Properties
  15. References (1/4)
  16. References (2/4)
  17. References (3/4)
  18. References (4/4)
  19. Index (1/3)
  20. Index (2/3)
  21. Index (3/3)

Product information

  • Title: Visualizing Quaternions
  • Author(s): Andrew J. Hanson
  • Release date: February 2006
  • Publisher(s): Elsevier Science
  • ISBN: None