Visualizing Quaternions

Andrew J. Hanson Ph.D. is an Emeritus Professor of Computer Science at Indiana University. He earned a bachelor’s degree in Chemistry and Physics from Harvard University in 1966 and a PhD in Theoretical Physics from MIT under Kerson Huang in 1971. His interests range from general relativity to computer graphics, artificial intelligence, and bioinformatics; he is particularly concerned with applications of quaternions and with exploitation of higher-dimensional graphics for the visualization of complex scientific contexts such as Calabi-Yau spaces. He is the co-discoverer of the Eguchi-Hanson “gravitational instanton” Einstein metric (1978), author of Visualizing Quaternions (Elsevier, 2006), and designer of the iPhone Apps “4Dice” and “4DRoom” (2012) for interacting with four-dimensional virtual reality.

• 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

“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