Pyramid Algorithms

A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling

By: Ron Goldman

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Hardcover | 10 July 2002

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Hardcover
576 Pages

Dimensions(cm)
24.13 x 19.68 x 3.18

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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

Industry Reviews

Ron Goldman is a leading expert who knows the fundamental concepts and their interconnectedness, as well as the small details. The elegance of the writing and of the methods used to present the material allows us to get a deep understanding of the central concepts of CAGD. In its simplicity and pure beauty, the theory indeed resembles the pyramids." --Helmut Pottman, Vienna University of Technology"A textbook approach to understanding, analyzing and computing common polynomial and spline curves, and surfaces schemes in computer-aided geometric modeling and design. Goldman employs a dynamic programming method based on recursive pyramids for revealing the structure and relationship of algorithms." --Design Issues

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You Can Find This Book In

Non-Fiction Computing & I.T.Computer Programming & Software Development Algorithms & Data Structures Graphical & Digital Media Applications Software Engineering Computer Science Mathematics Geometry

Foreword
Preface
Introduction: Foundations	p. 1
Ambient Spaces	p. 1
Coordinates	p. 27
Curve and Surface Representations	p. 38
Lagrange Interpolation and Neville's Algorithm	p. 47
Linear Interpolation	p. 47
Neville's Algorithm	p. 49
The Structure of Neville's Algorithm	p. 54
Uniqueness of Polynomial Interpolants and Taylor's Theorem	p. 56
Lagrange Basis Functions	p. 58
Computational Techniques for Lagrange Interpolation	p. 65
Rational Lagrange Curves	p. 69
Fast Fourier Transform	p. 77
Recapitulation	p. 83
Surface Interpolation	p. 84
Rectangular Tensor Product Lagrange Surfaces	p. 86
Triangular Lagrange Patches	p. 94
Uniqueness of the Bivariate Lagrange Interpolant	p. 103
Rational Lagrange Surfaces	p. 107
Ruled, Lofted, and Boolean Sum Surfaces	p. 111
Hermite Interpolation and the Extended Neville Algorithm	p. 119
Cubic Hermite Interpolation	p. 119
Neville's Algorithm for General Hermite Interpolation	p. 124
The Hermite Basis Functions	p. 130
Rational Hermite Curves	p. 135
Hermite Surfaces	p. 143
Newton Interpolation and Difference Triangles	p. 155
The Newton Basis	p. 156
Divided Differences	p. 157
Properties of Divided Differences	p. 165
An Axiomatic Approach to Divided Difference	p. 170
Forward Differencing	p. 173
Bezier Approximation and Pascal's Triangle	p. 187
De Casteljau's Algorithm	p. 188
Elementary Properties of Bezier Curves	p. 190
The Bernstein Basis Functions and Pascal's Triangle	p. 194
More Properties of Bernstein/Bezier Curves	p. 200
Change of Basis Procedures and Principles of Duality	p. 212
Differentiation and Integration	p. 238
Rational Bezier Curves	p. 255
Bezier Surfaces	p. 267
Blossoming	p. 307
Blossoming the de Casteljau Algorithm	p. 307
Existence and Uniqueness of the Blossom	p. 310
Change of Basis Algorithms	p. 317
Differentiation and the Homogeneous Blossom	p. 321
Blossoming Bezier Patches	p. 327
B-Spline Approximation and the de Boor Algorithm	p. 347
The de Boor Algorithm	p. 347
Progressive Polynomial Bases Generated by Progressive Knot Sequences	p. 355
B-Spline Curves	p. 358
Elementary Properties of B-Spline Curves	p. 361
All Splines Are B-Splines	p. 364
Knot Insertion Algorithms	p. 367
The B-Spline Basis Functions	p. 383
Uniform B-Splines	p. 405
Rational B-Splines	p. 418
Catmull-Rom Splines	p. 422
Tensor Product B-Spline Surfaces	p. 427
Pyramid Algorithms and Triangular B-Patches	p. 430
Pyramid Algorithms for Multisided Bezier Patches	p. 445
Barycentric Coordinates for Convex Polygons	p. 446
Polygonal Arrays	p. 450
Neville's Pyramid Algorithm and Multisided Grids	p. 454
S-Patches	p. 457
Pyramid Patches and the General Pyramid Algorithm	p. 473
C-Patches	p. 476
Toric Bezier Patches	p. 488
Index	p. 531
About the Author	p. 552
Table of Contents provided by Blackwell. All Rights Reserved.

Pyramid Algorithms

A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling

At a Glance

Hardcover

Industry Reviews

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How to return your order

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