Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry
- 11h 55m
- Daniel Fontijne, Leo Dorst, Stephen Mann
- Elsevier Science and Technology Books, Inc.
- 2007
Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming.
Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down.
Features
- Explains GA as a natural extension of linear algebra and conveys its significance for 3D programming of geometry in graphics, vision, and robotics.
- Systematically explores the concepts and techniques that are key to representing elementary objects and geometric operators using GA.
- Covers in detail the conformal model, a convenient way to implement 3D geometry using a 5D representation space.
- Presents effective approaches to making GA an integral part of your programming.
- Includes numerous drills and programming exercises helpful for both students and practitioners.
About the Authors
Leo Dorst is Assistant Professor of Computer Science at the University of Amsterdam, where his research focuses on geometrical issues in robotics and computer vision. He earned M.Sc. and Ph.D. degrees from Delft University of Technology and received a NYIPLA Inventor of the Year award in 2005 for his work in robot path planning.
Daniel Fontijne holds a Master's degree in artificial Intelligence and is a Ph.D. candidate in Computer Science at the University of Amsterdam. His main professional interests are computer graphics, motion capture, and computer vision.
Stephen Mann is Associate Professor in the David R. Cheriton School of Computer Science at the University of Waterloo, where his research focuses on geometric modeling and computer graphics. He has a B.A. in Computer Science and Pure Mathematics from the University of California, Berkeley, and a Ph.D. in Computer Science and Engineering from the University of Washington.
In this Book
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Why Geometric Algebra?
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Spanning Oriented Subspaces
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Metric Products of Subspaces
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Linear Transformations of Subspaces
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Intersection and Union of Subspaces
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The Fundamental Product of Geometric Algebra
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Orthogonal Transformations as Versors
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Geometric Differentiation
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Modeling Geometries
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The Vector Space Model: The Algebra of Directions
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The Homogeneous Model
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Applications of the Homogeneous Model
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The Conformal Model: Operational Euclidean Geometry
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New Primitives for Euclidean Geometry
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Constructions in Euclidean Geometry
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Conformal Operators
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Operational Models for Geometries
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Implementation Issues
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Basis Blades and Operations
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The Linear Products and Operations
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Fundamental Algorithms for Nonlinear Products
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Specializing the Structure for Efficiency
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Using the Geometry in a Ray-Tracing Application
