Visualizing Quaternions

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.

Autorentext

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.

Inhalt

About the Author; Preface; 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; References; Index