Geometric modeling with splines : an introduction

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000004802173	QA565 C65 2001	Open Access Book	Book	Searching... Unknown
Searching... PSZ KL	30000004802215	QA565 C65 2001	Open Access Book	Book	Searching... Unknown

Written by researchers who have helped found and shape the field, this book is a definitive introduction to geometric modeling. The authors present all of the necessary techniques for curve and surface representations in computer-aided modeling with a focus on how the techniques are used in design. They achieve a balance between mathematical rigor and broad applicability. Appropriate for readers with a moderate degree of mathematical maturity, this book is suitable as an undergraduate or graduate text, or particularly as a resource for self-study.

Author Notes

Cohen, Elaine; Riesenfeld, Richard F.; Elber, Gershon

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Cohen (Univ. of Utah), Riesenfeld (Univ. of Utah), and Elber (Technion, Israel Institute of Technology) offer a thorough introduction to the use of splines to approximate and represent curves and surfaces. The book is designed both as a text and a reference. It begins with a review of the linear algebra that is needed to understand the theory discussed in later chapters. Other basic topics such as conic sections and the Frenet equations are also discussed early on. Later sections cover Bezier curves and surfaces, B-splines (curves and surfaces), tensor product surfaces, and fitting curves and surfaces. The work is generally well written; definitions, theorems, and proofs are clearly presented, and there are also many helpful examples. The index is perhaps a bit brief considering the length of the work, but this is a minor point. Algorithms are represented in pseudo code; some 88 literature references and 98 exercises. Upper-division undergraduates through professionals. J. D. Fehribach Worcester Polytechnic Institute

I Introduction	p. 1
1 Review of Basic Concepts	p. 3
1.1 Vector Analysis	p. 3
1.1.1 R[superscript 2] and R[superscript 3] as Vector Spaces	p. 7
1.2 Linear Transformations	p. 12
1.3 Review of Matrix Properties	p. 14
1.4 Barycentric Coordinates	p. 18
1.5 Functions	p. 22
1.5.1 Equations of Lines	p. 24
1.5.2 Equations of Planes	p. 26
1.5.3 Polynomials	p. 29
1.5.4 Rational Functions	p. 32
1.6 Parametric or Vector Functions	p. 34
1.6.1 Function Characteristics	p. 37
II Curves	p. 43
2 Representation Issues	p. 45
2.1 The Other Representation--Data Set Representation	p. 46
2.2 Explicit Formulations	p. 47
2.3 Implicit Representation	p. 51
2.3.1 Differentiating Implicit Functions	p. 53
2.3.2 Graphing Implicit Functions	p. 54
2.4 Parametric Representation	p. 57
2.5 Shape Approximation	p. 67
3 Conic Sections	p. 73
3.1 The Use of Conic Sections in Design	p. 73
3.2 Locus of Points Definitions	p. 74
3.2.1 The Circle	p. 74
3.2.2 The Ellipse	p. 74
3.2.3 The Hyperbola	p. 76
3.2.4 The Parabola	p. 78
3.3 Conic Sections	p. 78
3.4 Implicit Quadratic Functions as Conics	p. 82
3.5 5 Point Construction	p. 88
3.6 Using Tangents	p. 91
3.6.1 Blending Formulation Revisited	p. 93
3.7 Conic Arcs as Rational Functions	p. 95
3.8 Piecewise Conic Sections	p. 106
3.9 Using Homogeneous Coordinates to Represent Conics	p. 107
3.9.1 Extending the Constructive Algorithm to All Conics	p. 107
3.9.2 The Homogeneous Coordinate System and the Projective Plane	p. 109
4 Differential Geometry for Space Curves	p. 113
4.1 More on Parameterizations	p. 113
4.2 Arc Length Parameterization	p. 115
4.3 Intrinsic Vectors of a Space Curve	p. 118
4.4 Frenet Equations	p. 121
4.5 Frenet Equations for Non-Arc Length Parameterizations	p. 127
4.6 Intrinsic Functions on Arbitrary Parameterizations	p. 127
4.7 Piecing together Parametric Curves	p. 131
4.8 Exercises	p. 134
5 Bezier Curves and Bernstein Approximation	p. 137
5.1 Constructive Evaluation Curves	p. 138
5.1.1 Derivatives of Constructive Evaluation Curves	p. 140
5.2 Bezier Curves	p. 142
5.2.1 Properties of Bernstein Blending Functions	p. 147
5.2.2 Curve Properties	p. 150
5.2.3 Derivative Evaluation	p. 151
5.2.4 Midpoint Subdivision	p. 154
5.3 Bernstein Approximation	p. 155
5.4 Interpolation Using the Bernstein Blending Functions	p. 160
5.4.1 Comparing Bezier Curves and Interpolation	p. 161
5.4.2 Comparing Bernstein Approximation to Functions and Interpolation	p. 162
5.5 Piecing together Bezier Curves	p. 164
5.6 Adding Flexibility--Degree Raising	p. 166
6 B-Spline Curves	p. 171
6.1 Constructive Piecewise Curves	p. 172
6.2 B-Spline Blending Functions	p. 183
6.2.1 Proof of Equivalence of Constructive Algorithm with Curve Form	p. 187
6.3 More Properties of B-Splines	p. 189
6.3.1 Continuity at the Curve Ends	p. 199
7 Linear Spaces of B-Splines	p. 205
7.1 Uniform Floating Spline Curves	p. 208
7.2 Spline Space Hierarchy	p. 209
7.2.1 Uniform Subdivison Curves	p. 213
7.3 Linear Independence of B-Splines	p. 218
7.4 A Second Look	p. 224
8 Choosing a B-Spline Space	p. 231
8.1 Knot Patterns	p. 231
8.1.1 Uniform vs. Nonuniform Knot Vectors	p. 235
8.1.2 End Conditions for B-Spline Curves	p. 236
8.1.3 Computing Uniform Floating B-Splines	p. 242
8.2 Analysis of Lower Degree Curves	p. 246
8.2.1 Quadratic B-Spline Curves	p. 247
8.2.2 Cubic B-Spline Curves	p. 250
8.3 Rational B-Spline Curves	p. 250
8.4 Effects of Multiple Knots vs. Multiple Vertices	p. 253
9 Data Fitting with B-Splines	p. 259
9.1 Interpolation with B-Splines	p. 259
9.1.1 C[superscript 2] Cubic Interpolation at Knots	p. 260
9.1.2 Higher-Order Complete Interpolation with B-Splines	p. 264
9.2 Other Positional Interpolation with B-Splines	p. 266
9.2.1 Nodal Interpolation	p. 267
9.2.2 Piecewise Cubic Hermite Interpolation	p. 268
9.2.3 Generalized Interpolation	p. 270
9.3 B-Spline Least Squares	p. 270
9.3.1 Direct Manipulation	p. 273
9.4 Schoenberg Variation Diminishing Splines	p. 274
9.5 Quasi-Interpolation	p. 279
9.5.1 Variation Diminishing Splines and B-Spline Curves	p. 283
9.6 Multiresolution Decompostion	p. 284
9.6.1 Multiresolution Curve Editing	p. 286
9.6.2 Constraints and Multiresolution Curve Editing	p. 288
10 Other Polynomial Bases for Interpolation	p. 291
10.1 Position Interpolation	p. 292
10.1.1 Hermite Interpolation	p. 295
10.2 Generalized Hermite Interpolation	p. 302
10.3 Piecewise Methods	p. 306
10.3.1 Piecewise Hermite Interpolation	p. 308
10.4 Parametric Extensions	p. 312
11 Other Derivations of B-Splines	p. 317
11.1 Higher-Dimensional Volumetric Projections	p. 317
11.2 Divided Difference Formulation	p. 322
11.2.1 Divided Differences	p. 322
11.2.2 Divided Differences for B-Splines	p. 328
11.3 Fast Evaluation of Polynomials Using Divided Differences	p. 332
11.3.1 Fast Calculations with Bezier Curves	p. 334
11.3.2 Fast Evaluation of B-Spline Curves	p. 334
11.4 Inverse Fourier Transform Definition of Splines	p. 335
III Surfaces	p. 339
12 Differential Geometry for Surfaces	p. 341
12.1 Regular Surfaces	p. 341
12.2 Tangents to Surfaces	p. 348
12.3 First Fundamental Form	p. 351
12.3.1 Arc Length of a Curve on a Simple Surface	p. 351
12.3.2 Invariance of the First Fundamental Form	p. 354
12.3.3 Angles between Tangent Vectors	p. 355
12.3.4 Surface Area of a Simple Surface	p. 356
12.4 The Second Fundamental Form and Curves in the Surface	p. 358
12.4.1 Examples	p. 368
12.4.2 The Osculating Paraboloid	p. 372
12.5 Surfaces	p. 377
13 Surface Representations	p. 381
13.1 Surface Representations	p. 381
13.2 Tensor Product Surfaces	p. 382
13.3 Evaluating Surfaces and Partial Derivatives and Rendering Isocurves	p. 390
13.3.1 Bezier Surface Evaluation	p. 393
13.3.2 Evaluation of Tensor Product B-Spline Surfaces	p. 396
13.4 Uniform Refinement for Uniform Floating Spline Surfaces	p. 399
13.4.1 Uniform Biquadratic Subdivision	p. 399
13.4.2 Uniform Bicubic Subdivision	p. 402
13.5 Matrix Expansions	p. 404
13.5.1 B-Splines and Bezier	p. 404
13.5.2 Bezier to Power Basis Evaluation Using Matrices	p. 405
13.5.3 Bicubic Hermite Surface Patches	p. 406
13.5.4 Comparing Coefficients in Bicubic Representations	p. 407
13.6 Other Polynomial-Based Surface Representations	p. 409
13.6.1 Simplex Splines	p. 409
13.6.2 Box Splines	p. 411
13.6.3 Bezier Triangles	p. 411
14 Fitting Surfaces	p. 415
14.1 Interpolating Tensor Product Surfaces	p. 415
14.2 Interpolation to a Grid of Data	p. 416
14.2.1 Nodal Interpolation	p. 417
14.2.2 Complete Cubic Spline Interpolation	p. 418
14.3 Surface Approximation	p. 422
14.3.1 Schoenberg Variation Diminishing Approximation	p. 422
14.3.2 Least Squares Surface Data Fitting	p. 422
14.4 Interpolating Arbitrary Boundary Curves: A Linear Operator Approach	p. 424
14.4.1 Coons Patches	p. 428
14.4.2 Using the Tensor Product Patch	p. 432
15 Modeling with B-Spline Surfaces	p. 435
15.1 Generalized Cylinder and Extrusion	p. 435
15.2 Ruled Surfaces	p. 437
15.3 Surface of Revolution	p. 438
15.3.1 The Cylinder	p. 439
15.3.2 The Cone	p. 440
15.3.3 The Truncated Cone	p. 440
15.4 Sweep Surfaces	p. 440
15.5 Normals and Offset Surfaces	p. 442
15.5.1 Given Four Sides	p. 443
15.5.2 Floating Surface	p. 445
15.6 Piecing together the Surfaces	p. 450
IV Advanced Techniques	p. 457
16 Subdivision and Refinement for Splines	p. 459
16.1 Refinement Theorems and Algorithms	p. 468
16.1.1 Special Case Algorithm	p. 475
16.1.2 Subdivision of Bezier Curves	p. 476
16.1.3 A More Efficient Algorithm	p. 480
16.2 Convergence of Refinement Strategies	p. 484
16.3 Degree Raising for Splines	p. 489
16.4 Restricted Proof for Bezier Subdivision	p. 492
17 Subdivision and Refinement in Modeling	p. 497
17.1 Subdivision of B-Spline Curves	p. 498
17.1.1 General Algorithm	p. 498
17.2 Curve Generation with Geometric Subdivision	p. 501
17.3 Curve-Curve Intersection	p. 503
17.4 Surface Refinement and Subdivision	p. 505
17.4.1 A Matrix Approach	p. 505
17.5 Rendering Open Surfaces	p. 511
17.5.1 Raster Images	p. 511
17.5.2 Line Drawings	p. 513
17.6 Using Subdivision and Refinement to Implement Design Operations	p. 514
17.6.1 Adding Local Degrees of Freedom	p. 514
18 Set Operations to Effect Modeling	p. 517
18.1 Intersections as Root Finding	p. 519
18.2 Lines	p. 520
18.3 Curve-Curve Intersections by Bisection	p. 520
18.4 Intersections with Newton-Raphson	p. 522
18.4.1 Curves	p. 522
18.4.2 Surfaces	p. 526
18.5 Intersections of Parametric Curves and Surfaces	p. 528
18.5.1 Parametric Curves	p. 528
18.5.2 Spline Intersections: Divide and Conquer	p. 528
18.5.3 Curve-Surface Intersections	p. 529
18.5.4 Ray-Surface Intersections	p. 531
18.5.5 Surface-Surface Intersection	p. 532
18.6 Boolean Operations on Models	p. 536
18.6.1 Boolean Operators by Membership Operators	p. 536
18.6.2 Evaluating Set Membership Functions	p. 539
18.6.3 Boolean Operators by Boundary Classification	p. 540
18.6.4 Computational Considerations in 2D Classification	p. 546
18.6.5 3D Classification	p. 552
18.7 Booleans on Sculptured Objects	p. 554
19 Model Data Structures	p. 557
19.1 The Winged-Edge Data Structure	p. 558
19.2 Trimmed Surfaces	p. 561
19.3 A Model	p. 563
19.4 Non-Manifold Geometry	p. 566
20 Subdivision Surfaces	p. 569
20.1 Catmull-Clark Subdivision Surfaces	p. 570
20.1.1 Subdivision Curves	p. 570
20.1.2 Regular Quad Mesh Surface Subdivision	p. 572
20.1.3 Arbitrary Grid Surface Subdivison	p. 574
20.1.4 Boundaries, Creases, and Darts	p. 577
20.1.5 Convergence	p. 581
20.2 Doo-Sabin Subdivision Surfaces	p. 582
20.2.1 Curves	p. 582
20.2.2 Regular Quad Mesh Surface Subdivision	p. 583
20.2.3 Arbitrary Grid Surface Subdivison	p. 584
20.2.4 Boundaries	p. 586
20.2.5 Convergence	p. 587
20.3 Triangular Subdivision Surfaces	p. 587
20.4 Other Types of Subdivision Surfaces	p. 589
21 Higher-Dimensional Tensor Product B-Splines	p. 591
21.1 Evaluation and Rendering of Trivariates	p. 592
21.2 Advanced Operations on Trivariates	p. 594
21.3 Constant Sets of Trivariates	p. 596
21.4 Construction of Trivariates	p. 597
21.4.1 Interpolation and Least Squares Fitting	p. 599
21.4.2 Traditional Constructors	p. 601
21.5 Warping Using Trivariates	p. 603
21.6 Varying Time	p. 604
Bibliography	p. 607
Index	p. 613

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