Visualizing Quaternions (The Morgan Kaufmann Series in Interactive 3D Technology) (英語) ハードカバー – 2006/1/12
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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.
“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
The book also serves as an accessible introduction into moving frames, which are the "right" way of treating the "extrinsic" differential geometry of curves and surfaces and which benefit a lot from a treatment using quaternions and spinors, leading to simpler equations. Some of the ideas presented here have been developed into a full-blown quaternionic framework for "extrinsic" surface geometry, by mathematicians of the Berlin school (U. Pinkall, A. Bobenko, etc.). The Caltech thesis of K. Crane ("Conformal Geometry Processing") might serve as a natural follow-up reading, for those whose interest has been piqued by Hanson's book.
It is also appreciated, that Hanson emphasizes the "square root" nature of quaternions, that's key to understand physical concepts like spin, the Dirac equation and supersymmetry, which are of tremendous importance in geometry and topology as well as theoretical physics, but may appear bizarre and abstract without some geometric intuition.
Part 1 is an introduction for those readers new to the topic. As far as introductions go, it is not too bad. It does in fact contain one important subject - quaternion interpolation - that is not always covered in other texts. Hanson covers interpolation in part 1 and again in part 2. If your interest is computer animation, this may be sufficient reason to acquire the book...analogous to purchasing an album just to get one song. However, if you are completely new to quaternions and want to develop a firm intuition grounded in first principles, then a book that is at least an order of magnitude better is "Quaternions and Rotation Sequences" by J. B. Kuipers.
Parts 2 and 3 are the most interesting parts of the book. Hanson presents a series of small chapters that discuss quaternions from different advanced mathematical viewpoints (differential geometry, group theory, Clifford algebras, octonions). The chapters are small, and so they by necessity contain references to the literature where the considerable background material required for understanding the topics is developed. If you have a good background in differential geometry and some abstract algebra, then the chapters are quite nice. In this sense, parts 2 and 3 of the book are more appropriate for mathematicians.
The technique of including routine, "turn the crank" type of calculations in the text, and deferring the sometimes considerable details and theory to references allows Hanson to cover more topics than usual. However, it is exactly those details that distinguish between what is useful and well conceived mathematical theory from mathematical gibberish. Deferring details to the literature can also encourage an over-reaching of the author beyond his understanding of the material. Hanson has walked a fine line here, but still I must mention two issues that I found annoying:
1) A Riemannian manifold is not specified only by giving the charts ("local patches") as Hanson seems to think on page 352. One must also add constraints on the topology -- typically Hausdorff with a countable basis of open sets. These are not just moot considerations; the topology allows a construction of a partition of unity which in turns guarantees the existence of the Riemannian metric. In particular, the mild condition of paracompactness will ensure the existence of the partition of unity.
2) It is a gross over-simplification, and mathematically non-trivial, to claim the basis vectors of Euclidean space have precise analogs in Fourier transform theory, as Hanson does on page 340. Heuristic analogs...yes... but precise analogs?...only if one has developed the necessary mathematical machinery using the theory of distributions. The inner product relation ei.ej = kronecker delta ij given by Hanson on page 340 would have to be generalized to a delta function. It was one of the major accomplishments of 20th century mathematics that Schwartz was able to put the delta function on a firm mathematical basis with his theory of distributions (for which he received the Fields medal) Before Schwartz, delta functions were at best a useful computational tool in the hands of physicists like Dirac who were guided by their physical intuition, and at worst, an example of the mathematical gibberish alluded to earlier.
In short, this is a good book for those with the mathematical prerequisites. Those with a more traditional background in computer science might be advised to first peruse a copy at their local bookstore to verify it matches their interests.