Pyramid Algorithms : A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling (Morgan Kaufmann Series in Computer Graphics)

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Pyramid Algorithms : A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling (Morgan Kaufmann Series in Computer Graphics)

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  • 製本 Hardcover:ハードカバー版/ページ数 551 p.
  • 言語 ENG
  • 商品コード 9781558603547
  • DDC分類 006.6

Full Description


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.

Table of Contents

Foreword                                           xiii
Preface xv
Introduction: Foundations 1 (44)
Ambient Spaces 1 (26)
Vector Spaces 1 (1)
Affine Spaces 2 (8)
Grassmann Spaces and Mass-Points 10 (7)
Projective Spaces and Points at Infinity 17 (4)
Mappings between Ambient Spaces 21 (3)
Polynomial and Rational Curves and 24 (3)
Surfaces
Coordinates 27 (11)
Rectangular Coordinates 28 (1)
Affine Coordinates, Grassmann 28 (3)
Coordinates, and Homogeneous Coordinates
Barycentric Coordinates 31 (7)
Curve and Surface Representations 38 (5)
Summary 43 (2)
Part I Interpolation 45 (140)
Lagrange Interpolation and Neville's Algorithm 47 (72)
Linear Interpolation 47 (2)
Neville's Algorithm 49 (5)
The Structure of Neville's Algorithm 54 (2)
Uniqueness of Polynomial Interpolants and 56 (2)
Taylor's Theorem
Lagrange Basis Functions 58 (7)
Computational Techniques for Lagrange 65 (4)
Interpolation
Rational Lagrange Curves 69 (8)
Fast Fourier Transform 77 (6)
Recapitulation 83 (1)
Surface Interpolation 84 (2)
Rectangular Tensor Product Lagrange Surfaces 86 (8)
Triangular Lagrange Patches 94 (9)
Uniqueness of the Bivariate Lagrange 103 (4)
Interpolant
Rational Lagrange Surfaces 107 (4)
Ruled, Lofted, and Boolean Sum Surfaces 111 (6)
Summary 117 (2)
Hermite Interpolation and the Extended 119 (36)
Neville Algorithm
Cubic Hermite Interpolation 119 (5)
Neville's Algorithm for General Hermite 124 (6)
Interpolation
The Hermite Basis Functions 130 (5)
Rational Hermite Curves 135 (8)
Hermite Surfaces 143 (11)
Tensor Product Hermite Surfaces 143 (5)
Lofted Hermite Surfaces 148 (2)
Boolean Sum Hermite Surfaces 150 (4)
Summary 154 (1)
Newton Interpolation and Difference Triangles 155 (30)
The Newton Basis 156 (1)
Divided Differences 157 (8)
Properties of Divided Differences 165 (5)
An Axiomatic Approach to Divided Difference 170 (3)
Forward Differencing 173 (7)
Summary 180 (5)
Identities for the Divided Difference 180 (5)
Part II Approximation 185 (346)
Bezier Approximation and Pascal's Triangle 187 (120)
De Casteljau's Algorithm 188 (2)
Elementary Properties of Bezier Curves 190 (4)
The Bernstein Basis Functions and Pascal's 194 (6)
Triangle
More Properties of Bernstein/Bezier Curves 200 (12)
Linear Independence and Nondegeneracy 200 (1)
Horner's Evaluation Algorithm for Bezier 201 (1)
Curves
Unimodality 202 (4)
Descartes' Law of Signs and the Variation 206 (6)
Diminishing Property
Change of Basis Procedures and Principles 212 (26)
of Duality
Conversion between Bezier and Monomial 217 (3)
Form
The Weierstrass Approximation Theorem 220 (4)
Degree Elevation for Bezier Curves 224 (5)
Subdivision 229 (1)
Sampling with Replacement 229 (2)
Subdivision Algorithm 231 (7)
Differentiation and Integration 238 (17)
Discrete Convolution and the Bernstein 239 (4)
Basis Functions
Differentiating Bernstein Polynomials and 243 (7)
Bezier Curves
Wang's Formula 250 (3)
Integrating Bernstein Polynomials and 253 (2)
Bezier Curves
Rational Bezier Curves 255 (12)
Differentiating Rational Bezier Curves 264 (3)
Bezier Surfaces 267 (30)
Tensor Product Bezier Patches 267 (12)
Triangular Bezier Patches 279 (14)
Rational Bezier Patches 293 (4)
Summary 297 (10)
Identities for the Bernstein Basis 299 (8)
Functions
Blossoming 307 (40)
Blossoming the de Casteljau Algorithm 307 (3)
Existence and Uniqueness of the Blossom 310 (7)
Change of Basis Algorithms 317 (4)
Differentiation and the Homogeneous Blossom 321 (6)
Blossoming Bezier Patches 327 (13)
Blossoming Triangular Bezier Patches 328 (7)
Blossoming Tensor Product Bezier Patches 335 (5)
Summary 340 (7)
Blossoming Identities 341 (6)
B-Spline Approximation and the de Boor 347 (98)
Algorithm
The de Boor Algorithm 347 (8)
Progressive Polynomial Bases Generated by 355 (3)
Progressive Knot Sequences
B-Spline Curves 358 (3)
Elementary Properties of B-Spline Curves 361 (3)
All Splines Are B-Splines 364 (3)
Knot Insertion Algorithms 367 (16)
Boehm's Knot Insertion Algorithm 368 (3)
The Oslo Algorithm 371 (4)
Change of Basis Algorithms via Knot 375 (1)
Insertion
Conversion to Piecewise Bezier Form 375 (1)
Bezier Subdivision and Conversion between 376 (3)
Bezier and Monomial Form
Differentiation and Knot Insertion 379 (1)
Differentiation as Knot Insertion 379 (1)
Boehm's Derivative Algorithm 380 (1)
Knot Insertion from Differentiation 381 (2)
The B-Spline Basis Functions 383 (22)
Elementary Properties of the B-Spline 386 (3)
Basis Functions
Blossoming and Dual Functionals 389 (2)
Differentiating and Integrating the 391 (3)
B-Splines
B-Splines and Divided Difference 394 (8)
A Geometric Characterization of the 402 (3)
B-Splines
Uniform B-Splines 405 (13)
Continuous Convolution and Uniform 406 (2)
B-Splines
Chaikin's Knot Insertion Algorithm 408 (3)
The Lane-Riesenfeld Knot Insertion 411 (7)
Algorithm
Rational B-Splines 418 (4)
Catmull-Rom Splines 422 (5)
Tensor Product B-Spline Surfaces 427 (3)
Pyramid Algorithms and Triangular B-Patches 430 (7)
Summary 437 (8)
Identities for the B-Spline Basis 439 (6)
Functions
Pyramid Algorithms for Multisided Bezier 445 (86)
Patches
Barycentric Coordinates for Convex Polygons 446 (4)
Polygonal Arrays 450 (4)
Neville's Pyramid Algorithm and Multisided 454 (3)
Grids
S-Patches 457 (16)
The Pyramid Algorithm and the S-Patch 459 (4)
Blending Functions
Simplicial S-Patches 463 (3)
Differentiating S-Patches 466 (3)
Blossoming S-Patches 469 (4)
Pyramid Patches and the General Pyramid 473 (3)
Algorithm
C-Patches 476 (12)
Toric Bezier Patches 488 (41)
Lattice Polygons 489 (2)
Barycentric Coordinates for Lattice 491 (4)
Polygons
The Pyramid Algorithm for Toric Bezier 495 (5)
Patches
The Boundaries of a Toric Bezier Patch 500 (3)
The Monomial and Bernstein 503 (2)
Representations of a Toric Bezier Patch
Toric S-Patches 505 (3)
Subdividing Toric Bezier Patches into 508 (9)
Tensor Product Bezier Patches
Depth Elevation for Toric Bezier Patches 517 (2)
Differentiating Toric Bezier Patches 519 (2)
Blossoming Toric Bezier Patches 521 (2)
Toric Bezier C-Patches 523 (6)
Summary 529 (2)
Index 531 (21)
About the Author 552