Subdivision Surfaces

by Peters, Jorg; Reif, Ulrich

• ISBN13: 9783540764052
• ISBN10: 3540764054
• Format: Hardcover
• Copyright: 2008-07-04
• Publisher: Springer Nature

Summary

Since their first appearance in 1974, subdivision algorithms for generating surfaces of arbitrary topology have gained widespread popularity in computer graphics and are being evaluated in engineering applications. This development was complemented by ongoing efforts to develop appropriate mathematical tools for a thorough analysis, and today, many of the fascinating properties of subdivision are well understood. This book summarizes the current knowledge on the subject. It contains both meanwhile classical results as well as brand-new, unpublished material, such as a new framework for constructing C^2-algorithms. The focus of the book is on the development of a comprehensive mathematical theory, and less on algorithmic aspects. It is intended to serve researchers and engineers - both new to the beauty of the subject - as well as experts, academic teachers and graduate students or, in short, anybody who is interested in the foundations of this flourishing branch of applied geometry.

Table of Contents

Introduction and Overview	p. 1
Refined Polyhedra	p. 1
Control Nets	p. 2
Splines with Singularities	p. 4
Focus and Scope	p. 6
Overview	p. 7
Notation	p. 7
Analysis in the Shift-Invariant Setting	p. 8
Historical Notes on Subdivision on Irregular Meshes	p. 11
Geometry Near Singularities	p. 15
Dot and Cross Products	p. 16
Regular Surfaces	p. 17
Surfaces with a Singular Point	p. 23
Criteria for Injectivity	p. 31
Generalized Splines	p. 39
An Alternative View of Spline Curves	p. 40
Continuous Bivariate Splines	p. 41
C[superscript k]-Splines	p. 44
C[subscript r superscript k]-Splines	p. 47
A Bicubic Illustration	p. 53
Subdivision Surfaces	p. 57
Refinability	p. 58
Segments and Rings	p. 59
Splines in Finite-Dimensional Subspaces	p. 65
Subdivision Algorithms	p. 67
Asymptotic Expansion of Sequences	p. 71
Jordan Decomposition	p. 72
The Subdivision Matrix	p. 75
C[subscript 1 superscript k]-Subdivision Algorithms	p. 83
Generic Initial Data	p. 84
Standard Algorithms	p. 84
General Algorithms	p. 89
Shift Invariant Algorithms	p. 95
Symmetric Algorithms	p. 103
Case Studies of C[subscript 1 superscript k]-Subdivision Algorithms	p. 109
Catmull-Clark Algorithm and Variants	p. 109
Doo-Sabin Algorithm and Variants	p. 116
Simplest Subdivision	p. 120
Shape Analysis and C[subscript 2 superscript k]-Algorithms	p. 125
Higher Order Asymptotic Expansions	p. 126
Shape Assessment	p. 134
Conditions for C[subscript 2 superscript k]-Algorithms	p. 140
A Framework for C[subscript 2 superscript k]-Algorithms	p. 145
Guided Subdivision	p. 149
Approximation and Linear Independence	p. 157
Proxy Splines	p. 157
Local and Global Linear Independence	p. 169
Conclusion	p. 175
Function Spaces	p. 176
Recursion	p. 177
Combinatorial Structure	p. 178
References	p. 183
Index	p. 199
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