Subdivision Methods for Geometric Design (eBook)

Front Cover 1
Subdivision Methods for Geometric Design: A Constructive Approach 4
Copyright Page 5
Contents 8
Foreword 6
Preface 12
Table of Symbols 14
Chapter 1. Subdivision: Functions as Fractals 18
1.1 Functions 18
1.2 Fractals 26
1.3 Subdivision 35
1.4 Overview 42
Chapter 2. An Integral Approach to Uniform Subdivision 44
2.1 A Subdivision Scheme for B-splines 45
2.2 A Subdivision Scheme for Box Splines 57
2.3 B-splines and Box Splines as Piecewise Polynomials 69
Chapter 3. Convergence Analysis for Uniform Subdivision Schemes 79
3.1 Convergence of a Sequence of Functions 80
3.2 Analysis of Univariate Schemes 86
3.3 Analysis of Bivariate Schemes 98
Chapter 4. A Differential Approach to Uniform Subdivision 108
4.1 Subdivision for B-splines 109
4.2 Subdivision for Box Splines 116
4.3 Subdivision for Exponential B-splines 120
4.4 A Smooth Subdivision Scheme with Circular Precision 127
Chapter 5. Local Approximation of Global Differential Schemes 137
5.1 Subdivision for Polyharmonic Splines 137
5.2 Local Approximations to Polyharmonic Splines 146
5.3 Subdivision for Linear Flows 158
Chapter 6. Variational Schemes for Bounded Domains 174
6.1 Inner Products for Stationary Subdivision Schemes 174
6.2 Subdivision for Natural Cubic Splines 184
6.3 Minimization of the Variational Scheme 197
6.4 Subdivision for Bounded Harmonic Splines 205
Chapter 7. Averaging Schemes for Polyhedral Meshes 215
7.1 Linear Subdivision for Polyhedral Meshes 215
7.2 Smooth Subdivision for Quad Meshes 221
7.3 Smooth Subdivision for Triangle Meshes 243
7.4 Other Types of Polyhedral Schemes 249
Chapter 8. Spectral Analysis at an Extraordinary Vertex 256
8.1 Convergence Analysis at an Extraordinary Vertex 256
8.2 Smoothness Analysis at an Extraordinary Vertex 266
8.3 Verifying the Smoothness Conditions for a Given Scheme 276
8.4 Future Trends in Subdivision 289
References 293
Index 304