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Tables of Contents for Pyramid Algorithms
Chapter/Section Title
Page #
Page Count
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 Surfaces
24
3
Coordinates
27
11
Rectangular Coordinates
28
1
Affine Coordinates, Grassmann Coordinates, and Homogeneous Coordinates
28
3
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 Taylor's Theorem
56
2
Lagrange Basis Functions
58
7
Computational Techniques for Lagrange Interpolation
65
4
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 Interpolant
103
4
Rational Lagrange Surfaces
107
4
Ruled, Lofted, and Boolean Sum Surfaces
111
6
Summary
117
2
Hermite Interpolation and the Extended Neville Algorithm
119
36
Cubic Hermite Interpolation
119
5
Neville's Algorithm for General Hermite Interpolation
124
6
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 Triangle
194
6
More Properties of Bernstein/Bezier Curves
200
12
Linear Independence and Nondegeneracy
200
1
Horner's Evaluation Algorithm for Bezier Curves
201
1
Unimodality
202
4
Descartes' Law of Signs and the Variation Diminishing Property
206
6
Change of Basis Procedures and Principles of Duality
212
26
Conversion between Bezier and Monomial Form
217
3
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 Basis Functions
239
4
Differentiating Bernstein Polynomials and Bezier Curves
243
7
Wang's Formula
250
3
Integrating Bernstein Polynomials and Bezier Curves
253
2
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 Functions
299
8
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 Algorithm
347
98
The de Boor Algorithm
347
8
Progressive Polynomial Bases Generated by Progressive Knot Sequences
355
3
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 Insertion
375
1
Conversion to Piecewise Bezier Form
375
1
Bezier Subdivision and Conversion between Bezier and Monomial Form
376
3
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 Basis Functions
386
3
Blossoming and Dual Functionals
389
2
Differentiating and Integrating the B-Splines
391
3
B-Splines and Divided Difference
394
8
A Geometric Characterization of the B-Splines
402
3
Uniform B-Splines
405
13
Continuous Convolution and Uniform B-Splines
406
2
Chaikin's Knot Insertion Algorithm
408
3
The Lane-Riesenfeld Knot Insertion Algorithm
411
7
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 Functions
439
6
Pyramid Algorithms for Multisided Bezier Patches
445
86
Barycentric Coordinates for Convex Polygons
446
4
Polygonal Arrays
450
4
Neville's Pyramid Algorithm and Multisided Grids
454
3
S-Patches
457
16
The Pyramid Algorithm and the S-Patch Blending Functions
459
4
Simplicial S-Patches
463
3
Differentiating S-Patches
466
3
Blossoming S-Patches
469
4
Pyramid Patches and the General Pyramid Algorithm
473
3
C-Patches
476
12
Toric Bezier Patches
488
41
Lattice Polygons
489
2
Barycentric Coordinates for Lattice Polygons
491
4
The Pyramid Algorithm for Toric Bezier Patches
495
5
The Boundaries of a Toric Bezier Patch
500
3
The Monomial and Bernstein Representations of a Toric Bezier Patch
503
2
Toric S-Patches
505
3
Subdividing Toric Bezier Patches into Tensor Product Bezier Patches
508
9
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