CONTENTS CHAPTER 1. WHY GEOMETRIC ALGEBRA? An Example in Geometric Algebra How It Works and How It’s Different Vector Spaces as Modeling Tools Subspaces as Elements of Computation Linear Transformations Extended Universal Orthogonal Transformations Objects are Operators Closed-Form Interpolation and Perturbation Programming Geometry You Can Only Gain Software Implementation The Structure of This Book Part I: Geometric Algebra Part II: Models of Geometry Part III: Implementation of Geometric Algebra The Structure of the Chapters PART I GEOMETRIC ALGEBRA CHAPTER 2. SPANNING ORIENTED SUBSPACES Vector Spaces Oriented Line Elements Properties of Homogeneous Lines Visualizing Vectors Oriented Area Elements Properties of Planes Introducing the Outer Product Visualizing Bivectors Visualizing Bivector Addition Oriented Volume Elements Properties of Volumes Associativity of the Outer Product Visualization of Trivectors Quadvectors in 3-D Are Zero Scalars Interpreted Geometrically Applications Solving Linear Equations Intersecting Planar Lines Homogeneous Subspace Representation Parallelness Direct Representation of Oriented Weighted Subspaces Nonmetric Lengths, Areas, and Volumes Th
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.
The first book on Geometric Algebra for programmers in computer graphics and entertainment computing
Written by leaders in the field providing essential information on this new technique for 3D graphics
This full colour book includes a website with GAViewer, a program to experiment with GA
Professionals working in fields requiring complex geometric computation such as robotics, computer graphics, and computer games. Students in graduate or advanced undergraduate programs in computer science.
- No. of pages:
- © Morgan Kaufmann 2007
- 19th April 2007
- Morgan Kaufmann
- Hardcover ISBN:
Informatics Institute, Faculty of Sciences, University of Amsterdam, The Netherlands
Daniel Fontijne holds a Master’s degree in artificial Intelligence and a Ph.D. in Computer Science, both from the University of Amsterdam. His main professional interests are computer graphics, motion capture, and computer vision.
Intelligent Autonomous Systems, University of Amsterdam, The Netherlands
University of Waterloo, Ontario, Canada