Geometric algebra for computer science an object-oriented approach to geometry
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 requ...
| Otros Autores: | , , |
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| Formato: | Libro electrónico |
| Idioma: | Inglés |
| Publicado: |
Amsterdam, [Netherlands] :
Morgan Kaufmann Publishers
2007.
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| Colección: | Morgan Kaufmann series in computer graphics.
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| Materias: | |
| Ver en Biblioteca Universitat Ramon Llull: | https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009634738106719 |
Tabla de Contenidos:
- 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
