Geometric algebra for computer science : an object-oriented approach to geometry / Leo Dorst, Daniel Fontijne, Stephen Mann.

Format:

Book

Author/Creator:

Dorst, Leo, 1958- author.

Fontijne, Daniel, author.

Mann, Stephen, 1963- author.

Series:

Morgan Kaufmann series in computer graphics.

The Morgan Kaufmann series in computer graphics

Language:

English

Subjects (All):

Geometry, Algebraic.

Computer science--Mathematics.

Computer science.

Object-oriented methods (Computer science).

Physical Description:

1 online resource (663 p.)

Place of Publication:

Amsterdam, [Netherlands] : Morgan Kaufmann Publishers, 2007.

Language Note:

English

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 limit

Contents:

Front Cover; Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry; Copyright Page; Contents; LIST OF FIGURES; LIST OF TABLES; LIST OF PROGRAMMING EXAMPLES; PREFACE; CHAPTER 1. WHY GEOMETRIC ALGEBRA?; 1.1 An Example in Geometric Algebra; 1.2 How It Works and How It's Different; 1.3 Programming Geometry; 1.4 The Structure of This Book; 1.5 The Structure of the Chapters; 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 Zero2.6 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 L; 3.5 Orthogonality and Duality; 3.6 Orthogonal Projection of Subspaces; 3.7 The 3-D Cross Product

3.8 Application: Reciprocal Frames3.9 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 Factorization5.3 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; 6.1 The Geometric Product for Vectors; 6.2 The Geometric Product of Multivectors; 6.3 The Subspace Products Retrieved; 6.4 Geometric Division; 6.5 Further Reading; 6.6 Exercises

6.7 Programming Examples and ExercisesCHAPTER 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; 8.1 Geometrical Changes by Orthogonal Transformations; 8.2 Transformational Changes; 8.3 Parametric Differentiation

8.4 Scalar Differentiation

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-282-25807-9

9786612258077

0-08-095879-6

OCLC:

460116023