Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling

577 Pages·2002·22.12 MB·English

by  Ron Goldman

Pyramid Algorithms
A Dynamic Programming Approach to
Cu rves and Surfaces for Geometric Modeling
The Morgan Kaufmann Series in Computer Graphics and
Geometric Modeling
Series Editor: Brian A. Barsky, University of California, Berkeley
Geometric Tools for Computer Graphics Andrew Glassner's Notebook:
Philip Schneider and David Eberly Recreational Computer Graphics
Texturing and Modeling: A Procedural Andrew S. Glassner
Approach, Third Edition Warping and Morphing of Graphical Objects
David S. Ebert, E Kenton Musgrave, Darwyn Jonas Gomes, Lucia Darsa, Bruno Costa, and
Peachey, Ken Perlin, and Steven Worley Luiz Velho
Jim Blinn's Corner: Notation, Notation, Jim Blinn's Corner: Dirty Pixels
Notation Jim Blinn
Jim Blinn Rendering with Radiance: The Art and
Understanding Virtual Reality Science of Lighting Visualization
William Sherman and Alan Craig Greg Ward Larson and Rob Shakespeare
Level of Detail for 3D Graphics: Introduction to Implicit Surfaces
Application and Theory Edited by Jules Bloomenthal
David Luebke, Martin Reddy, Jonathan D. Jim Blinn's Corner: A Trip Down
Cohen, Amitabh Varshney, Benjamin Watson, the Graphics Pipeline
and Robert Huebner Jim Blinn
Digital Video and HDTV Algorithms and Interactive Curves and Surfaces:
Interfaces A Multimedia Tutorial on CAGD
Charles Poynton Alyn Rockwood and Peter Chambers
Pyramid Algorithms: A Dynamic Wavelets for Computer Graphics:
Programming Approach to Curves and Theory and Applications
Surfaces for Geometric Modeling Eric J. Stollnitz, Tony D. DeRose, and
Ron Goldman David H. Salesin
Non-Photorealistic Computer Graphics: Principles of Digital Image Synthesis
Modeling, Rendering, and Animation
Andrew S. Glassner
Thomas Strothotte and Stefan Schlechtweg
Radiosity & Global Illumination
Curves and Surfaces for CAGD : A Practical
Frangois X. Sillion and Claude Puech
Guide, Fifth Edition
Knotty: A B-Spline Visualization Program
Gerald Farin
Jonathan Yen
Subdivision Methods for Geometric Design:
User Interface Management Systems:
A Constructive Approach
Models and Algorithms
Joe Warren and Henrik Weimer
Dan R. Olsen, Jr.
Computer Animation: Algorithms and
Making Them Move: Mechanics, Control, and
Techniques
Animation of Articulated Figures
Rick Parent
Edited by Norman I. Badler, Brian A. Barsky,
The Computer Animator's Technical Handbook
and David Zeltzer
Lynn Pocock and Judson Rosebush
Geometric and Solid Modeling: An
Advanced RenderMan: Introduction
Creating CGl for Motion Pictures
Christoph M. Hoffmann
Anthony A. Apodaca and Larry Gritz
An Introduction to Splines for Use in Computer
Curves and Surfaces in Geometric Modeling: Graphics and Geometric Modeling
Theory and Algorithms
Richard H. Bartels, John C. Beatty, and
Jean Gallier Brian A. Barsky
Pyramid Algorithms
A Dynamic Programming Approach to
Curves and Surfaces for Geometric Modeling
Ron Goldman
Rice University
M ~4 ~
MORGAN KAUFMANN PUBLISHERS
AN IMPRINT OF ELSEVIER SCIENCE
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Morgan Kaufmann Publishers
An Imprint of Elsevier Science
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(cid:14)9 2003 by Elsevier Science (USA)
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written permission of the publisher.
Library of Congress Control Number: 2002107240
ISBN 1-55860-354-9
This book is printed on acid-free paper.
For my graduate students~past, present, and future
This writer.., would have to prepare his book with meticulous care,
perpetually regrouping his forces like a general conducting an offen-
sive, and he would have also to endure his book like a form of
fatigue, to accept it like a discipline, build it up like a church, follow
it like a medical regime, vanquish it like an obstacle, win it like a
friendship, cosset it like a little child, create it like a new world with-
out neglecting those mysteries whose explanation is to be found
probably only in worlds other than our own and the presentiment of
which is the thing that moves us most deeply in life and in art.
Marcel Proust, Remembrance of Things Past
This Page Intentionally Left Blank
Contents
Foreword xiii
Preface XV
Chapter 1 Introduction: Foundations
1.1 Ambient Spaces
1 1.1 Vector Spaces 1
1 1.2 Affine Spaces 2
1 1.3 Grassmann Spaces and Mass-Points 10
1 1.4 Projective Spaces and Points at Infinity 17
1 1.5 Mappings between Ambient Spaces 21
1 1.6 Polynomial and Rational Curves and Surfaces 24
1.2 Coordinates 27
1.2.1 Rectangular Coordinates 28
1.2.2 Affine Coordinates, Grassmann Coordinates, and
Homogeneous Coordinates 28
1.2.3 Barycentric Coordinates 31
1.3 Curve and Surface Representations 38
1.4 Summary 43
Part I Interpolation 45
Chapter 2 Lagrange Interpolation and Neville's Algorithm 47
2.1 Linear Interpolation 47
2.2 Neville's Algorithm 49
2.3 The Structure of Neville's Algorithm 54
2.4 Uniqueness of Polynomial Interpolants and Taylor's Theorem 56
2.5 Lagrange Basis Functions 58
2.6 Computational Techniques for Lagrange Interpolation 65
2.7 Rational Lagrange Curves 69
vii
viii Contents
2.8 FastF ourier Transform 77
2.9 Recapitulation 83
2.10 Surface Interpolation 84
2.11 Rectangular Tensor Product Lagrange Surfaces 86
2.12 Triangular Lagrange Patches 94
2.13 Uniqueness of the Bivariate Lagrange Interpolant 103
214 Rational Lagrange Surfaces 107
2.15 Ruled, Lofted, and Boolean Sum Surfaces 111
2.16 Summary 117
Chapter 3 Hermite Interpolation and the Extended Neville Algorithm 119
3.1 Cu bic Hermite Interpolation 119
3.2 Neville'sA lgorithm for General Hermite Interpolation 124
3.3 The Hermite Basis Functions 130
3.4 Rational Hermite Curves 135
3.5 HermiteS urfaces 143
3.5.1 TensorP roduct Hermite Surfaces 143
3.5.2 Lofted Hermite Surfaces 148
3.5.3 Boolean Sum Hermite Surfaces 150
3.6 Summary 154
Chapter 4 Newton Interpolation and Difference Triangles 155
4.1 The Newton Basis 15 6
4.2 DividedD ifferences 15 7
4.3 Propertieso f Divided Differences 165
4.4 An Axiomatic Approach to Divided Difference 170
4.5 ForwardD ifferencing 173
4.6 Summary 180
4.6.1 Identities for the Divided Difference 180
Part II Approximation 185
Chapter 5 Bezier Approximation and Pascal's Triangle 187
5.1 De Casteljau's Algorithm 188
5.2 Elementary Properties of Bezier Curves 190
5.3 The Bernstein Basis Functions and Pascal's Triangle 194
5.4 More Properties of Bernstein/Bezier Curves 200
5.4.1 Linear Independence and Nondegeneracy 200
5.4.2 Horner's Evaluation Algorithm for Bezier Curves 201
5.4.3 Unimodality 202
5.4.4 Descartes' Law of Signs and the Variation Diminishing Property 206
Contents ix
5.5 Change of Basis Procedures and Principles of Duality 212
5.5.1 Conversion between Bezier and Monomial Form 217
5.5.2 The Weierstrass Approximation Theorem 220
5.5.3 Degree Elevation for Bezier Curves 224
5.5.4 Subdivision 229
5.5.4.1 Sampling with Replacement 229
5.5.4.2 SubdivisionA lgorithm 231
5.6 Differentiation and Integration 238
5.6.1 DiscreteC onvolution and the Bernstein Basis Functions 239
5.6.2 Differentiating Bernstein Polynomials and Bezier Curves 243
5.6.3 Wang's Formula 250
5.6.4 Integrating Bernstein Polynomials and Bezier Curves 253
5.7 Rational Bezier Curves 255
5.7.1 Differentiating Rational Bezier Curves 264
5.8 Bezier Surfaces
267
5.8.1 Tensor Product Bezier Patches 267
5.8.2 Triangular Bezier Patches 279
5.8.3 Rational Bezier Patches 293
5.9 Summary 297
5.9.1 Identitiesf or the Bernstein Basis Functions 299
Chapter 6 Blossoming 307
6.1 Blossomingt he de Casteljau Algorithm 307
6.2 Existencea nd Uniqueness of the Blossom 310
6.3 Change of Basis Algorithms 317
6.4 Differentiation and the Homogeneous Blossom 321
6.5 Blossoming Bezier Patches 327
6.5.1 BlossomingT riangular Bezier Patches 328
6.5.2 Blossoming Tensor Product Bezier Patches 335
6.6 Summary 340
6.6.1 Blossoming Identities 341
Chapter 7 B-Spline Approximation and the de Boor Algorithm 347
7.1 The de Boor Algorithm 347
7.2 ProgressiveP olynomial Bases Generated by Progressive
Knot Sequences 355
7.3 B-SplineC urves 358
7.4 Elementary Properties of B-Spline Curves 361
7.5 All SplinesAre B-Splines 364
7.6 Knot Insertion Algorithms 367
7.6.1 Boehm's Knot Insertion Algorithm 368