Geometric Algebra for Computer Science (Revised Edition)

by Dorst; Fontijne; Mann

• ISBN13: 9780123749420
• ISBN10: 0123749425
• Format: Hardcover
• Copyright: 2009-03-23
• Publisher: Elsevier Science

Summary

Until recently, 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.*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. *Companion web site includes links to GAViewer, a program that will allow you to interact with many of the 3D figures in the book, and Gaigen 2, the platform for the instructive programming exercises that conclude each chapter.

Table of Contents

Why Geometric Algebra?

Geometric Algebra

Spanning Oriented Subspaces

Metric Products Of Subspaces

Linear Transformations Of Subspaces

Intersection And Union Of Subspaces

The Fundamental Product Of Geometric Algebra

Orthogonal Transformations As Versors

Geometric Differentiation

Models Of Geometries

Modeling Geometries

The Vector Space Model: The Algebra Of Directions

The Homogeneous Model

Applications Of The Homogeneous Model

The Conformal Model: Operational Euclidean Geometry

New Primitives For Euclidean Geometry

Constructions In Euclidean Geometry

Conformal Operators

Operational Models For Geometries

Implementing Geometric Algebra

Implementation Issues

Basis Blades And Operations

The Linear Products And Operations

Fundamental Algorithms For Nonlinear Products

Specializing The Structure For Efficiency

Using The Geometry In A Ray- Tracing Application

Appendices A Metrics And Null Vectors

Contractions And Other Inner Products

Subspace Products Retrieved

Common Equations

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.