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Visualizing Quaternions

  • ID: 3025343
  • Book
  • 600 Pages
  • Elsevier Science and Technology
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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.
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About the Author
I Elements of Quaternions
1 The Discovery of Quaternions
2 Rotations Take the Stage
3 Basic Notation
4 What Are Quaternions?
5 Roadmap to Quaternion Visualization
6 Basic Rotations
7 Visualizing Algebraic Structure
8 Visualizing Quaternion Spheres
9 Visualizing Logarithms and Exponentials
10 Basic Interpolation Methods
11 Logarithms and Exponentials for Rotations
12 Seeing Elementary Quaternion Frames
13 Quaternions and the Belt Trick
14 More about the Rolling Ball: Order-Dependence is Good
15 More About Gimbal Lock
II Advanced Quaternion Applications and Topics
16 Alternative Ways to Write Down Quaternions
17 Efficiency and Complexity Issues
18 Advanced Sphere Visualization
19 Orientation Frames and Rotations
20 Quaternion Frame Methods
21 Quaternion Curves and Surfaces
22 Quaternion Curves
23 Quaternion Surfaces
24 Quaternion Volumes
25 Quaternion Maps of Streamlines and Flow Fields
26 Quaternion Interpolation
27 Controlling Quaternion Animation
28 Global Minimization: Advanced Interpolation
29 Quaternion Rotator Dynamics
30 Spherical Riemann Geometry
31 Quaternion Barycentric Coordinates
32 Quaternions and Representations of the Rotation Group
33 Quaternions and the Four Division Algebras
34 Clifford Algebras
35 Conclusion
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 Evolver
H The Relationship of 4D Rotations to Quaternions
I Quaternion Frame Integration
J Hyperspherical Geometry
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Hanson, Andrew J.
Andrew J. Hanson is a professor of computer science at Indiana University in Bloomington, Indiana, and has taught courses in computer graphics, computer vision, programming languages, and scientific visualization. He received a BA in chemistry and physics from Harvard College and a PhD in theoretical physics from MIT. Before coming to Indiana University, he did research in theoretical physics at the Institute for Advanced Study, Cornell University, the Stanford Linear Accelerator Center, and the Lawrence-Berkeley Laboratory, and then in computer vision at the SRI Artificial Intelligence Center in Menlo Park, CA. He has published a wide variety of technical articles concerning problems in theoretical physics, machine vision, computer graphics, and scientific visualization methods. His current research interests include scientific visualization (with applications in mathematics, cosmology and astrophysics, special and general relativity, and string theory), optimal model selection, machine vision, computer graphics, perception, collaborative methods in virtual reality, and the design of interactive user interfaces for virtual reality and visualization applications.
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