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ISBN 10: 0080553109
ISBN 13: 9780080553108
Author: Leo Dorst, Daniel Fontijne, Stephen Mann
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.
Chapter 1: Why Geometric Algebra?
1.1 An Example in Geometric Algebra
1.2 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
1.3 Programming Geometry
1.4 The Structure of This Book
Part I: Geometric Algebra
Part II: Models of Geometry
1.5 The Structure of the Chapters
Exercises
Drills
Structural Exercises
Programming Examples and Exercises
Part I: Geometric Algebra
Chapter 2: Spanning Oriented Subspaces
2.1 Vector Spaces
2.2 Oriented Line Elements
Visualizing Vectors
Oriented Area Elements
Properties of Planes
2.3 Introducing the Outer Product
Visualizing Bivectors
Visualizing Bivector Addition
2.4 Oriented Volume Elements
Properties of Volumes
Associativity of the Outer Product
Visualization of Trivectors
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
Exercises
Drills
Structural Exercises
Programming Examples and Exercises
Chapter 3: Meet and Join: Projecting and Intersecting Subspaces
3.1 Duality and the Pseudoscalar
3.2 The Right Contraction
From Scalar Product to Contraction
Implicit Definition of Contraction
Computing the Contraction Explicitly
Algebraic Subtleties
Geometric Interpretation of the Contraction
The Other Contraction
Orthogonality and Duality
Nonassociativity of the Contraction
3.3 The Inverse of a Blade
3.4 Orthogonal Complement and Duality
The Duality Relationships
Dual Representation of Subspaces
Orthogonal Projection of Subspaces
3.5 The 3-D Cross Product
Use of the Cross Product
The Cross Product Incorporated
3.6 Application: Reciprocal Frames
3.7 Further Reading
Exercises
Drills
Structural Exercises
Programming Examples and Exercises
Chapter 4: The Geometric Product: The Glue of Geometric Algebra
4.1 The Geometric Product
Product of Vector and Scalar
Product of Vector and Itself
Product of Two Perpendicular Vectors
Product of Two Arbitrary Vectors
Geometric Product of Any Two Vectors
4.2 Inner and Outer Product from Geometric Product
4.3 Product of a Vector and a Blade
Reversion
4.4 The Full Geometric Product: Any Two Multivectors
Associativity
Inverses
Commutativity
4.5 The Inverse of a Multivector
4.6 Further Reading
Exercises
Drills
Structural Exercises
Programming Examples and Exercises
Chapter 5: Transformations: Operators in Geometric Algebra
5.1 Rotations: Spinors
Reflection in a Hyperplane
Rotation as Two Reflections
The Rotor as an Operator
Properties of Rotors
Rotors for Other Planes
Spinor
Rotation of a Vector
5.2 General Orthogonal Transformations
Inverses of Outermorphisms
Matrix Representations
Matrices for Vector Transformations
Matrices for Outermorphisms
5.3 Summary
5.4 Suggestions for Further Reading
Exercises
Structural Exercises
Programming Examples and Exercises
Chapter 6: Metrics: How to Measure Things
6.1 What Is a Metric?
6.2 Orthogonal Bases
6.3 Euclidean Space
6.4 Non-Euclidean Metrics
6.5 Further Reading
Exercises
Drills
Structural Exercises
Programming Examples and Exercises
Chapter 7: Differentiations: Constructing Functionals and Operators
7.1 The Vector Derivative
7.2 The Geometric Derivative
7.3 Vector Fields and Differential Operators
7.4 The Cauchy-Riemann Equations
7.5 Further Reading
Exercises
Structural Exercises
Programming Examples and Exercises
Part II: Models of Geometry
Chapter 8: The Standard Model of Euclidean Geometry: 3D Euclidean Space
8.1 Basic Objects
8.2 Basic Operators
8.3 Distances and Angles
8.4 Projections
8.5 Intersections
8.6 Examples
8.7 Further Reading
Exercises
Structural Exercises
Programming Examples and Exercises
Chapter 9: The Homogeneous Model: 3D Euclidean Space with Points at Infinity
9.1 Introduction to the Homogeneous Model
9.2 Points and Euclidean Subspaces
9.3 General Euclidean Subspaces
9.4 Orthogonal Transformations (Rotations and Translations)
9.5 Reflections
9.6 Projections and Rejections
9.7 Intersections
9.8 Applications
9.9 Further Reading
Exercises
Structural Exercises
Programming Examples and Exercises
Chapter 10: The Conformal Model: Circles, Spheres, and Beyond
10.1 Introduction to the Conformal Model
10.2 Points in the Conformal Model
10.3 Geometric Primitives as Blades
Spheres
Planes
Circles
Lines
Point Pairs
10.4 Transformations in the Conformal Model
Rotations
Translations
Dilations (Scaling)
Inversions
10.5 Projective Transformations
10.6 The Meet and Join Operations
10.7 Distances
10.8 Projections
10.9 Intersection of Geometric Primitives
10.10 Applications
10.11 Further Reading
Exercises
Structural Exercises
Programming Examples and Exercises
Chapter 11: Implementations of Geometric Algebra
11.1 Designing an Effective GA Implementation
11.2 Generic GA Implementations
11.3 Code Generators
11.4 Optimized Implementations
11.5 Examples of Implementations
11.6 Further Reading
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Tags: Leo Dorst, Daniel Fontijne, Stephen Mann, Geometric, Algebra
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