Goldman Pyramid Algorithms

A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling
1. Auflage 2002
ISBN: 978-0-08-051547-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 577 Seiten

Pyramid Algorithms presents a unique approach to understanding, analyzing, and computing the most common polynomial and spline curve and surface schemes used in computer-aided geometric design, employing a dynamic programming method based on recursive pyramids.
The recursive pyramid approach offers the distinct advantage of revealing the entire structure of algorithms, as well as relationships between them, at a glance. This book-the only one built around this approach-is certain to change the way you think about CAGD and the way you perform it, and all it requires is a basic background in calculus and linear algebra, and simple programming skills.
* Written by one of the world's most eminent CAGD researchers
* Designed for use as both a professional reference and a textbook, and addressed to computer scientists, engineers, mathematicians, theoreticians, and students alike
* Includes chapters on Bezier curves and surfaces, B-splines, blossoming, and multi-sided Bezier patches
* Relies on an easily understood notation, and concludes each section with both practical and theoretical exercises that enhance and elaborate upon the discussion in the text
* Foreword by Professor Helmut Pottmann, Vienna University of Technology

Ron Goldman is a researcher at Sun Microsystems Laboratories in California working on alternative software development methodologies and new software architectures inspired by biology. He has been working with open source since hacking on GDB at Lucid, Inc. back in 1992. Since 1998 he has been helping groups at Sun Microsystems understand open source and advising them on how to build successful communities around their open source projects.
Prior to Sun he developed a program to generate and manipulate visual representations of complex data for use by social scientists as part of a collaboration between NYNEX Science & Technology and the Institute for Research on Learning. He has worked on programming language design, programming environments, user interface design, and data visualization. He has a PhD in computer science from Stanford University where he was a member of the robotics group.

Autoren/Hrsg.

Goldman, Ron

Inhaltsverzeichnis

1;Front Cover;1
2;Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling;4
3;Copyright Page;5
4;Contents;8
5;Foreword;14
6;Preface;16
7;Chapter 1. Introduction: Foundations;26
7.1;1.1 Ambient Spaces;26
7.2;1.2 Coordinates;52
7.3;1.3 Curve and Surface Representations;63
7.4;1.4 Summary;68
8;Part I: Interpolation;70
8.1;Chapter 2. Lagrange Interpolation and Neville's Algorithm;72
8.1.1;2.1 Linear Interpolation;72
8.1.2;2.2 Neville's Algorithm;74
8.1.3;2.3 The Structure of Neville's Algorithm;79
8.1.4;2.4 Uniqueness of Polynomial Interpolants and Taylor's Theorem;81
8.1.5;2.5 Lagrange Basis Functions;83
8.1.6;2.6 Computational Techniques for Lagrange Interpolation;90
8.1.7;2.7 Rational Lagrange Curves;94
8.1.8;2.8 Fast Fourier Transform;102
8.1.9;2.9 Recapitulation;108
8.1.10;2.10 Surface Interpolation;109
8.1.11;2.11 Rectangular Tensor Product Lagrange Surfaces;111
8.1.12;2.12 Triangular Lagrange Patches;119
8.1.13;2.13 Uniqueness of the Bivariate Lagrange Interpolant;128
8.1.14;214 Rational Lagrange Surfaces;132
8.1.15;2.15 Ruled, Lofted, and Boolean Sum Surfaces;136
8.1.16;2.16 Summary;142
8.2;Chapter 3. Hermite Interpolation and the Extended Neville Algorithm;144
8.2.1;3.1 Cubic Hermite Interpolation;144
8.2.2;3.2 Neville's Algorithm for General Hermite Interpolation;149
8.2.3;3.3 The Hermite Basis Functions;155
8.2.4;3.4 Rational Hermite Curves;160
8.2.5;3.5 Hermite Surfaces;168
8.2.6;3.6 Summary;179
8.3;Chapter 4. Newton Interpolation and Difference Triangles;180
8.3.1;4.1 The Newton Basis;181
8.3.2;4.2 Divided Differences;182
8.3.3;4.3 Properties of Divided Differences;190
8.3.4;4.4 An Axiomatic Approach to Divided Difference;195
8.3.5;4.5 Forward Differencing;198
8.3.6;4.6 Summary;205
9;Part II: Approximation;210
9.1;Chapter 5. Bezier Approximation and Pascal's Triangle;212
9.1.1;5.1 De Casteljau's Algorithm;213
9.1.2;5.2 Elementary Properties of Bezier Curves;215
9.1.3;5.3 The Bernstein Basis Functions and Pascal's Triangle;219
9.1.4;5.4 More Properties of Bernstein/Bezier Curves;225
9.1.5;5.5 Change of Basis Procedures and Principles of Duality;237
9.1.6;5.6 Differentiation and Integration;263
9.1.7;5.7 Rational Bezier Curves;280
9.1.8;5.8 Bezier Surfaces;292
9.1.9;5.9 Summary;322
9.2;Chapter 6. Blossoming;332
9.2.1;6.1 Blossoming the de Casteljau Algorithm;332
9.2.2;6.2 Existence and Uniqueness of the Blossom;335
9.2.3;6.3 Change of Basis Algorithms;342
9.2.4;6.4 Differentiation and the Homogeneous Blossom;346
9.2.5;6.5 Blossoming Bezier Patches;352
9.2.6;6.6 Summary;365
9.3;Chapter 7. B-Spline Approximation and the de Boor Algorithm;372
9.3.1;7.1 The de Boor Algorithm;372
9.3.2;7.2 Progressive Polynomial Bases Generated by Progressive Knot Sequences;380
9.3.3;7.3 B-Spline Curves;383
9.3.4;7.4 Elementary Properties of B-Spline Curves;386
9.3.5;7.5 All Splines Are B-Splines;389
9.3.6;7.6 Knot Insertion Algorithms;392
9.3.7;7.7 The B-Spline Basis Functions;408
9.3.8;7.8 Uniform B-Splines;430
9.3.9;7.9 Rational B-Splines;443
9.3.10;7.10 Catmull-Rom Splines;447
9.3.11;7.11 Tensor Product B-Spline Surfaces;452
9.3.12;7.12 Pyramid Algorithms and Triangular B-Patches;455
9.3.13;7.13 Summary;462
9.4;Chapter 8. Pyramid Algorithms for Multisided Bezier Patches;470
9.4.1;8.1 Barycentric Coordinates for Convex Polygons;471
9.4.2;8.2 Polygonal Arrays;475
9.4.3;8.3 Neville's Pyramid Algorithm and Multisided Grids;479
9.4.4;8.4 S-Patches;482
9.4.5;8.5 Pyramid Patches and the General Pyramid Algorithm;498
9.4.6;8.6 C-Patches;501
9.4.7;8.7 Toric Bezier Patches;513
9.4.8;8.8 Summary;554
10;Index;556
11;About the Author;577