Book description
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
Table of contents
- Cover
- Title Page
- Copyright
- List of Figures
- List of Tables
- List of Programming Examples
- Preface
- Table of Contents
- Chapter 1: Why geometric algebra?
-
Part I: Geometric algebra
-
Chapter 2: Spanning oriented subspaces
- 2.1 Vector Spaces
- 2.2 Oriented Line Elements
- 2.3 Oriented Area Elements
- 2.4 Oriented Volume Elements
- 2.5 Quadvectors in 3-D are Zero
- 2.6 Scalars Interpreted Geometrically
- 2.7 Applications
- 2.8 Homogeneous Subspace Representation
- 2.9 The Graded Algebra of Subspaces
- 2.10 Summary of Outer Product Properties
- 2.11 Further Reading
- 2.12 Exercises
- 2.13 Programming Examples and Exercises
-
Chapter 3: Metric products of subspaces
- 3.1 Sizing Up Subspaces
- 3.2 From Scalar Product to Contraction
- 3.3 Geometric Interpretation of the Contraction
- 3.4 The Other Contraction ⌊
- 3.5 Orthogonality and Duality
- 3.6 Orthogonal Projection of Subspaces
- 3.7 The 3-D Cross Product
- 3.8 Application: Reciprocal Frames
- 3.9 Further Reading
- 3.10 Exercises
- 3.11 Programming Examples and Exercises
-
Chapter 4: Linear transformations of subspaces
- 4.1 Linear Transformations of Vectors
- 4.2 Outermorphisms: Linear Transformations of Blades
- 4.3 Linear Transformation of the Metric Products
- 4.4 Inverses of Outermorphisms
- 4.5 Matrix Representations
- 4.6 Summary
- 4.7 Suggestions For Further Reading
- 4.8 Structural Exercises
- 4.9 Programming Examples and Exercises
-
Chapter 5: Intersection and union of subspaces
- 5.1 The Phenomenology of Intersection
- 5.2 Intersection Through Outer Factorization
- 5.3 Relationships Between Meet and Join
- 5.4 Using Meet and Join
- 5.5 Join and Meet are Mostly Linear
- 5.6 Quantitative Properties of the Meet
- 5.7 Linear Transformation of Meet and Join
- 5.8 Offset Subspaces
- 5.9 Further Reading
- 5.10 Exercises
- 5.11 Programming Examples and Exercises
- Chapter 6: The fundamental product of geometric algebra
-
Chapter 7: Orthogonal transformations as versors
- 7.1 Reflections of Subspaces
- 7.2 Rotations of Subspaces
- 7.3 Composition of Rotations
- 7.4 The Exponential Representation of Rotors
- 7.5 Subspaces As Operators
- 7.6 Versors Generate Orthogonal Transformations
- 7.7 The Product Structure of Geometric Algebra
- 7.8 Further Reading
- 7.9 Exercises
- 7.10 Programming Examples and Exercises
- Chapter 8: Geometric differentiation
-
Chapter 2: Spanning oriented subspaces
-
Part II: Models of geometries
- Chapter 9: Modeling geometries
- Chapter 10: The vector space model
-
Chapter 11: The homogeneous model
- 11.1 Homogeneous Representation Space
- 11.2 All Points are Vectors
- 11.3 All Lines are 2-Blades
- 11.4 All Planes are 3-Blades
- 11.5 k-Flats as (k + 1)-Blades
- 11.6 Direct and Dual Representations of Flats
- 11.7 Incidence Relationships
- 11.8 Linear Transformations: Motions and More
- 11.9 Coordinate-Free Parameterized Constructions
- 11.10 Metric Products in the Homogeneous Model
- 11.11 Further Reading
- 11.12 Exercises
- 11.13 Programming Examples and Exercises
- Chapter 12: Applications of the homogeneous model
-
Chapter 13: The conformal model
- 13.1 The Conformal Model
- 13.2 Euclidean Transformations As Versors
- 13.3 Flats and Directions
- 13.4 Application: General Planar Reflection
- 13.5 Rigid Body Motions
- 13.6 Application: Interpolation of Rigid Body Motions
- 13.7 Application: Differential Planar Reflections
- 13.8 Further Reading
- 13.9 Exercises
- 13.10 Programming Examples and Exercises
- Chapter 14: New primitives for euclidean geometry
- Chapter 15: Constructions in euclidean geometry
- Chapter 16: Conformal operators
- Chapter 17: Operational models for geometries
-
Part III: Implementing geometric algebra
- Chapter 18: Implementation issues
-
Chapter 19: Basis blades and operations
- 19.1 Representing Unit Basis Blades With Bitmaps
- 19.2 The Outer Product of Basis Blades
- 19.3 The Geometric Product of Basis Blades in An Orthogonal Metric
- 19.4 The Geometric Productofbasis Blades in Nonorthogonal Metrics
- 19.5 The Metric Products of Basis Blades
- 19.6 Commutator Product of Basis Blades
- 19.7 Grade-Dependent Signs On Basis Blades
- Chapter 20: The linear products and operations
- Chapter 21: Fundamental algorithms for nonlinear products
- Chapter 22: Specializing the structure for efficiency
- Chapter 23: Using the geometry in a ray-tracing application
- Metrics and null vectors
- Contractions and other inner products
- Subspace products retrieved
- Common equations
- Bibliography
- Index
- Instructions for Online Access
Product information
- Title: Geometric Algebra for Computer Science
- Author(s):
- Release date: July 2010
- Publisher(s): Morgan Kaufmann
- ISBN: 9780080553108
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