Geometric modeling with splines: an introduction
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Natick, Mass.
Peters
2001
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 616 S. graph. Darst. |
| ISBN: | 1568811373 |
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| 245 | 1 | 0 | |a Geometric modeling with splines |b an introduction |c Elaine Cohen ; Richard F. Riesenfeld ; Gershon Elber |
| 264 | 1 | |a Natick, Mass. |b Peters |c 2001 | |
| 300 | |a XXII, 616 S. |b graph. Darst. | ||
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| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Computer-aided design | |
| 650 | 4 | |a Curves on surfaces |x Mathematical models | |
| 650 | 4 | |a Spline theory | |
| 650 | 4 | |a Surfaces |x Mathematical models | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
I Introduction 1
1 Review of Basic Concepts 3
1.1 Vector Analysis 3
1.1.1 R2 and R3 as Vector Spaces 7
1.2 Linear Transformations 12
1.3 Review of Matrix Properties 14
1.4 Barycentric Coordinates 18
1.5 Functions 22
1.5.1 Equations of Lines 24
1.5.2 Equations of Planes 26
1.5.3 Polynomials 29
1.5.4 Rational Functions 32
1.6 Parametric or Vector Functions 34
1.6.1 Function Characteristics 37
II Curves 43
2 Representation Issues 45
2.1 The Other Representation—Data Set Representation .... 46
2.2 Explicit Formulations 47
2.3 Implicit Representation 51
2.3.1 Differentiating Implicit Functions 53
2.3.2 Graphing Implicit Functions 54
xv
xvi Contents
2.4 Parametric Representation 57
2.5 Shape Approximation 67
3 Conic Sections 73
3.1 The Use of Conic Sections in Design 73
3.2 Locus of Points Definitions 74
3.2.1 The Circle 74
3.2.2 The Ellipse 74
3.2.3 The Hyperbola 76
3.2.4 The Parabola 78
3.3 Conic Sections 78
3.4 Implicit Quadratic Functions as Conies 82
3.5 5 Point Construction 88
3.6 Using Tangents 91
3.6.1 Blending Formulation Revisited 93
3.7 Conic Arcs as Rational Functions 95
3.8 Piecewise Conic Sections 106
3.9 Using Homogeneous Coordinates to Represent Conies . . . 107
3.9.1 Extending the Constructive Algorithm to All Conies 107
3.9.2 The Homogeneous Coordinate System and
the Projective Plane 109
4 Differential Geometry for Space Curves 113
4.1 More on Parameterizations 113
4.2 Arc Length Parameterization 115
4.3 Intrinsic Vectors of a Space Curve 118
4.4 Frenet Equations 121
4.5 Frenet Equations for Non Arc Length Parameterizations . . 127
4.6 Intrinsic Functions on Arbitrary Parameterizations 127
4.7 Piecing together Parametric Curves 131
4.8 Exercises 134
5 Bezier Curves and Bernstein Approximation 137
5.1 Constructive Evaluation Curves 138
5.1.1 Derivatives of Constructive Evaluation Curves . . . 140
5.2 Bezier Curves 142
5.2.1 Properties of Bernstein Blending Functions 147
5.2.2 Curve Properties 150
5.2.3 Derivative Evaluation 151
5.2.4 Midpoint Subdivision 154
5.3 Bernstein Approximation 155
Contents xvii
5.4 Interpolation Using the Bernstein Blending Functions . . . 160
5.4.1 Comparing Bezier Curves and Interpolation 161
5.4.2 Comparing Bernstein Approximation to
Functions and Interpolation 162
5.5 Piecing together Bezier Curves 164
5.6 Adding Flexibility—Degree Raising 166
6 B Spline Curves 171
6.1 Constructive Piecewise Curves 172
6.2 B Spline Blending Functions 183
6.2.1 Proof of Equivalence of Constructive Algorithm with
Curve Form 187
6.3 More Properties of B Splines 189
6.3.1 Continuity at the Curve Ends 199
7 Linear Spaces of B Splines 205
7.1 Uniform Floating Spline Curves 208
7.2 Spline Space Hierarchy 209
7.2.1 Uniform Subdivison Curves 213
7.3 Linear Independence of B Splines 218
7.4 A Second Look 224
8 Choosing a B Spline Space 231
8.1 Knot Patterns 231
8.1.1 Uniform vs. Nonuniform Knot Vectors 235
8.1.2 End Conditions for B Spline Curves 236
8.1.3 Computing Uniform Floating B Splines 242
8.2 Analysis of Lower Degree Curves 246
8.2.1 Quadratic B Spline Curves 247
8.2.2 Cubic B Spline Curves 250
8.3 Rational B Spline Curves 250
8.4 Effects of Multiple Knots vs. Multiple Vertices 253
9 Data Fitting with B Splines 259
9.1 Interpolation with B Splines 259
9.1.1 C2 Cubic Interpolation at Knots 260
9.1.2 Higher Order Complete Interpolation with B Splines 264
9.2 Other Positional Interpolation with B Splines 266
9.2.1 Nodal Interpolation 267
9.2.2 Piecewise Cubic Hermite Interpolation 268
9.2.3 Generalized Interpolation 270
xviii Contents
9.3 B Spline Least Squares 270
9.3.1 Direct Manipulation 273
9.4 Schoenberg Variation Diminishing Splines 274
9.5 Quasi Interpolation 279
9.5.1 Variation Diminishing Splines and B Spline Curves . 283
9.6 Multiresolution Decompostion 284
9.6.1 Multiresolution Curve Editing 286
9.6.2 Constraints and Multiresolution Curve Editing . . . 288
10 Other Polynomial Bases for Interpolation 291
10.1 Position Interpolation 292
10.1.1 Hermite Interpolation 295
10.2 Generalized Hermite Interpolation 302
10.3 Piecewise Methods 306
10.3.1 Piecewise Hermite Interpolation 308
10.4 Parametric Extensions 312
11 Other Derivations of B Splines 317
11.1 Higher Dimensional Volumetric Projections 317
11.2 Divided Difference Formulation 322
11.2.1 Divided Differences 322
11.2.2 Divided Differences for B Splines 328
11.3 Fast Evaluation of Polynomials Using Divided Differences . 332
11.3.1 Fast Calculations with Bezier Curves 334
11.3.2 Fast Evaluation of B Spline Curves 334
11.4 Inverse Fourier Transform Definition of Splines 335
III Surfaces 339
12 Differential Geometry for Surfaces 341
12.1 Regular Surfaces 341
12.2 Tangents to Surfaces 348
12.3 First Fundamental Form 351
12.3.1 Arc Length of a Curve on a Simple Surface 351
12.3.2 Invariance of the First Fundamental Form 354
12.3.3 Angles between Tangent Vectors 355
12.3.4 Surface Area of a Simple Surface 356
12.4 The Second Fundamental Form and Curves in the Surface . 358
12.4.1 Examples 368
12.4.2 The Osculating Paraboloid 372
12.5 Surfaces 377
Contents xix
13 Surface Representations 381
13.1 Surface Representations 381
13.2 Tensor Product Surfaces 382
13.3 Evaluating Surfaces and Partial Derivatives and
Rendering Isocurves 390
13.3.1 Bezier Surface Evaluation 393
13.3.2 Evaluation of Tensor Product B Spline Surfaces ... 396
13.4 Uniform Refinement for Uniform Floating Spline Surfaces . 399
13.4.1 Uniform Biquadratic Subdivision 399
13.4.2 Uniform Bicubic Subdivision 402
13.5 Matrix Expansions 404
13.5.1 B Splines and Bezier 404
13.5.2 Bezier to Power Basis Evaluation Using Matrices . . 405
13.5.3 Bicubic Hermite Surface Patches 406
13.5.4 Comparing Coefficients in Bicubic Representations . 407
13.6 Other Polynomial Based Surface Representations 409
13.6.1 Simplex Splines 409
13.6.2 Box Splines 411
13.6.3 Bezier Triangles 411
14 Fitting Surfaces 415
14.1 Interpolating Tensor Product Surfaces 415
14.2 Interpolation to a Grid of Data 416
14.2.1 Nodal Interpolation 417
14.2.2 Complete Cubic Spline Interpolation 418
14.3 Surface Approximation 422
14.3.1 Schoenberg Variation Diminishing Approximation . 422
14.3.2 Least Squares Surface Data Fitting 422
14.4 Interpolating Arbitrary Boundary Curves:
A Linear Operator Approach 424
14.4.1 Coons Patches 428
14.4.2 Using the Tensor Product Patch 432
15 Modeling with B Spline Surfaces 435
15.1 Generalized Cylinder and Extrusion 435
15.2 Ruled Surfaces 437
15.3 Surface of Revolution 438
15.3.1 The Cylinder 439
15.3.2 The Cone 440
15.3.3 The Truncated Cone 440
xx Contents
15.4 Sweep Surfaces 440
15.5 Normals and Offset Surfaces 442
15.5.1 Given Four Sides 443
15.5.2 Floating Surface 445
15.6 Piecing together the Surfaces 450
IV Advanced Techniques 457
16 Subdivision and Refinement for Splines 459
16.1 Refinement Theorems and Algorithms 468
16.1.1 Special Case Algorithm 475
16.1.2 Subdivision of Bezier Curves 476
16.1.3 A More Efficient Algorithm 480
16.2 Convergence of Refinement Strategies 484
16.3 Degree Raising for Splines 489
16.4 Restricted Proof for Bezier Subdivision 492
17 Subdivision and Refinement in Modeling 497
17.1 Subdivision of B Spline Curves 498
17.1.1 General Algorithm 498
17.2 Curve Generation with Geometric Subdivision 501
17.3 Curve Curve Intersection 503
17.4 Surface Refinement and Subdivision 505
17.4.1 A Matrix Approach 505
17.5 Rendering Open Surfaces 511
17.5.1 Raster Images 511
17.5.2 Line Drawings 513
17.6 Using Subdivision and Refinement to Implement
Design Operations 514
17.6.1 Adding Local Degrees of Freedom 514
18 Set Operations to Effect Modeling 517
18.1 Intersections as Root Finding 519
18.2 Lines 520
18.3 Curve Curve Intersections by Bisection 520
18.4 Intersections with Newton Raphson 522
18.4.1 Curves 522
18.4.2 Surfaces 526
18.5 Intersections of Parametric Curves and Surfaces 528
18.5.1 Parametric Curves 528
18.5.2 Spline Intersections: Divide and Conquer 528
Contents xxi
18.5.3 Curve Surface Intersections 529
18.5.4 Ray Surface Intersections 531
18.5.5 Surface Surface Intersection 532
18.6 Boolean Operations on Models 536
18.6.1 Boolean Operators by Membership Operators .... 536
18.6.2 Evaluating Set Membership Functions 539
18.6.3 Boolean Operators by Boundary Classification . . . 540
18.6.4 Computational Considerations in 2D Classification . 546
18.6.5 3D Classification 552
18.7 Booleans on Sculptured Objects 554
19 Model Data Structures 557
19.1 The Winged Edge Data Structure 558
19.2 Trimmed Surfaces 561
19.3 A Model 563
19.4 Non Manifold Geometry 566
20 Subdivision Surfaces 569
20.1 Catmull Clark Subdivision Surfaces 570
20.1.1 Subdivision Curves 570
20.1.2 Regular Quad Mesh Surface Subdivision 572
20.1.3 Arbitrary Grid Surface Subdivison 574
20.1.4 Boundaries, Creases, and Darts 577
20.1.5 Convergence 581
20.2 Doo Sabin Subdivision Surfaces 582
20.2.1 Curves 582
20.2.2 Regular Quad Mesh Surface Subdivision 583
20.2.3 Arbitrary Grid Surface Subdivison 584
20.2.4 Boundaries 586
20.2.5 Convergence 587
20.3 Triangular Subdivision Surfaces 587
20.4 Other Types of Subdivision Surfaces 589
21 Higher Dimensional Tensor Product B Splines 591
21.1 Evaluation and Rendering of Trivariates 592
21.2 Advanced Operations on Trivariates 594
21.3 Constant Sets of Trivariates 596
21.4 Construction of Trivariates 597
21.4.1 Interpolation and Least Squares Fitting 599
21.4.2 Traditional Constructors 601
21.5 Warping Using Trivariates 603
21.6 Varying Time 604
«ii Contents
Bibliography 607
Index 613
|
| any_adam_object | 1 |
| author | Cohen, Elaine Riesenfeld, Richard F. Elber, Gershon |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:58:02Z |
| institution | BVB |
| isbn | 1568811373 |
| language | English |
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| spelling | Cohen, Elaine Verfasser aut Geometric modeling with splines an introduction Elaine Cohen ; Richard F. Riesenfeld ; Gershon Elber Natick, Mass. Peters 2001 XXII, 616 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Conception assistée par ordinateur Courbes sur les surfaces - Modèles mathématiques Geometrische methoden gtt Krommen gtt Oppervlakken gtt Representatie (wiskunde) gtt Splines gtt Splines, Théorie des Surfaces - Modèles mathématiques Mathematisches Modell Computer-aided design Curves on surfaces Mathematical models Spline theory Surfaces Mathematical models Geometrische Modellierung (DE-588)4156717-1 gnd rswk-swf Spline (DE-588)4182391-6 gnd rswk-swf Spline (DE-588)4182391-6 s Geometrische Modellierung (DE-588)4156717-1 s DE-604 Riesenfeld, Richard F. Verfasser aut Elber, Gershon Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009675757&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cohen, Elaine Riesenfeld, Richard F. Elber, Gershon Geometric modeling with splines an introduction Conception assistée par ordinateur Courbes sur les surfaces - Modèles mathématiques Geometrische methoden gtt Krommen gtt Oppervlakken gtt Representatie (wiskunde) gtt Splines gtt Splines, Théorie des Surfaces - Modèles mathématiques Mathematisches Modell Computer-aided design Curves on surfaces Mathematical models Spline theory Surfaces Mathematical models Geometrische Modellierung (DE-588)4156717-1 gnd Spline (DE-588)4182391-6 gnd |
| subject_GND | (DE-588)4156717-1 (DE-588)4182391-6 |
| title | Geometric modeling with splines an introduction |
| title_auth | Geometric modeling with splines an introduction |
| title_exact_search | Geometric modeling with splines an introduction |
| title_full | Geometric modeling with splines an introduction Elaine Cohen ; Richard F. Riesenfeld ; Gershon Elber |
| title_fullStr | Geometric modeling with splines an introduction Elaine Cohen ; Richard F. Riesenfeld ; Gershon Elber |
| title_full_unstemmed | Geometric modeling with splines an introduction Elaine Cohen ; Richard F. Riesenfeld ; Gershon Elber |
| title_short | Geometric modeling with splines |
| title_sort | geometric modeling with splines an introduction |
| title_sub | an introduction |
| topic | Conception assistée par ordinateur Courbes sur les surfaces - Modèles mathématiques Geometrische methoden gtt Krommen gtt Oppervlakken gtt Representatie (wiskunde) gtt Splines gtt Splines, Théorie des Surfaces - Modèles mathématiques Mathematisches Modell Computer-aided design Curves on surfaces Mathematical models Spline theory Surfaces Mathematical models Geometrische Modellierung (DE-588)4156717-1 gnd Spline (DE-588)4182391-6 gnd |
| topic_facet | Conception assistée par ordinateur Courbes sur les surfaces - Modèles mathématiques Geometrische methoden Krommen Oppervlakken Representatie (wiskunde) Splines Splines, Théorie des Surfaces - Modèles mathématiques Mathematisches Modell Computer-aided design Curves on surfaces Mathematical models Spline theory Surfaces Mathematical models Geometrische Modellierung Spline |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009675757&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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