Visualizing Quaternions

A fresh look at a classic technique in computer graphics and game development

Andrew J. Hanson (Author), Steve Cunningham (Foreword by)

9780120884001, Elsevier Science

Hardback, published 6 February 2006

536 pages, Approx. 125 Illustrations
23.4 x 19 x 3.1 cm, 1.27 kg

“Almost all computer graphics practitioners have a good grasp of the 3D Cartesian space. However, in many graphics applications, orientations and rotations are equally important, and the concepts and tools related to rotations are less well-known.
Quaternions are the key tool for understanding and manipulating orientations and rotations, and this book does a masterful job of making quaternions accessible. It excels not only in its scholarship, but also provides enough detailed figures and examples to expose the subtleties encountered when using quaternions. This is a book our field has needed for twenty years and I’m thrilled it is finally here.? --Peter Shirley, Professor, University of Utah

“This book contains all that you would want to know about quaternions, including a great many things that you don’t yet realize that you want to know!? --Alyn Rockwood, Vice President, ACM SIGGRAPH

“We need to use quaternions any time we have to interpolate orientations, for animating a camera move, simulating a rollercoaster ride, indicating fluid vorticity or displaying a folded protein, and it’s all too easy to do it wrong. This book presents gently but deeply the relationship between orientations in 3D and the differential geometry of the three-sphere in 4D that we all need to understand to be proficient in modern science and engineering, and especially computer graphics.? --John C. Hart, Associate Professor, Department of Computer Science, University of Illinois Urbana-Champaign, and Editor-in-Chief, ACM Transactions on Graphics

“Visualizing Quaternions is a comprehensive, yet superbly readable introduction to the concepts, mechanics, geometry, and graphical applications of Hamilton’s lasting contribution to the mathematical description of the real world. To write effectively on this subject, an author has to be a mathematician, physicist and computer scientist; Hanson is all three.
Still, the reader can afford to be much less learned since the patient and detailed explanations makes this book an easy read.? --George K. Francis, Professor, Mathematics Department, University of Illinois at Urbana-Champaign

“The new book, Visualizing Quaternions, will be welcomed by the many fans of Andy Hanson’s SIGGRAPH course.? --Anselmo Lastra, University of North Carolina at Chapel Hill

“Andy Hanson’s expository yet scholarly book is a stunning tour de force; it is both long overdue, and a splendid surprise! Quaternions have been a perennial source of confusion for the computer graphics community, which sorely needs this book. His enthusiasm for and deep knowledge of the subject shines through his exceptionally clear prose, as he weaves together a story encompassing branches of mathematics from group theory to differential geometry to Fourier analysis. Hanson leads the reader through the thicket of interlocking mathematical frameworks using visualization as the path, providing geometric interpretations of quaternion properties.
The first part of the book features a lucid explanation of how quaternions work that is suitable for a broad audience, covering such fundamental application areas as handling camera trajectories or the rolling ball interaction model. The middle section will inform even a mathematically sophisticated audience, with careful development of the more subtle implications of quaternions that have often been misunderstood, and presentation of less obvious quaternion applications such as visualizing vector field streamlines or the motion envelope of the human shoulder joint. The book concludes with a bridge to the mathematics of higher dimensional analogues to quaternions, namely octonians and Clifford algebra, that is designed to be accessible to computer scientists as well as mathematicians.? --Tamara Munzner, University of British Columbia

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.

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

Subject Areas: Graphics programming [UML], Computer games / online games: strategy guides [UDX], Applied mathematics [PBW], Algebra [PBF]