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.

Autorentext
Andrew J. Hanson 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.

Klappentext
The night the first Mars Exploration Rover landed, the engineering team at the Jet Propulsion Laboratory in Pasadena excitedly announced: "We have excellent quaternions for the rover!" This meant that through the application of a technique developed 160 years before they were able to determine the precise orientation of the spacecraft on Mars after traveling millions of miles through space. Quaternions are widely used in computer graphics, simulations, Hollywood special effects, computer games, and space travel, but they are conceptually very hard to grasp. The computer graphics community has a general familiarity with quaternions and their uses, but there are still many unresolved issues and controversies with them. And the subject mystifies many new engineers, such as game developers who lack a background in mathematics. While there are books that approach quaternions from a mathematician's perspective, this is the first that will give a professional developer an intuitive understanding of quaternions through visual representations. Visualizing Quaternions introduces new developers to this powerful technique and will provide the definitive explanation to those who have been struggling to use them. 1558600210LangtextGame development is still in its infancy, but games are poised for a major evolution: Ready to move from mostly plot-driven stories to more character based ones. The problem is that the principles of sophisticated character design and interaction are not widely understood in the game development community. And gender and cultural issues within the game genre are growing in importance. Katherine Isbister has spent 10 years examining what makes interactions with computer characters useful and engaging to different audiences. The key is to understand what's memorable, exciting, useful, and engaging to the participant about real-life social interactions, and applying that to character design. Game designers who create great characters often make use of these principles without realizing it. This book gives game design professionals and other designers a framework for how social roles and perceptual manipulations function, so that they are more aware of what they are doing when they design characters, and why it works.

Zusammenfassung
"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 …

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