Mathematical Methods in Computer Aided Geometric Design II (eBook)

Tom Lyche, Larry L. Schumaker (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
644 Seiten
Elsevier Science (Verlag)
978-1-4832-5798-3 (ISBN)

Infos

Mathematical Methods in Computer Aided Geometric Design II covers the proceedings of the 1991 International Conference on Curves, Surfaces, CAGD, and Image Processing, held at Biri, Norway. This book contains 48 chapters that include the topics of blossoming, cyclides, data fitting and interpolation, and finding intersections of curves and surfaces. Considerable chapters explore the geometric continuity, geometrical optics, image and signal processing, and modeling of geological structures. The remaining chapters discuss the principles of multiresolution analysis, NURBS, offsets, radial basis functions, rational splines, robotics, spline and Bézier methods for curve and surface modeling, subdivision, terrain modeling, and wavelets. This book will prove useful to mathematicians, computer scientists, and advance mathematics students.

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Mathematical Methods in Computer Aided Geometric Design II covers the proceedings of the 1991 International Conference on Curves, Surfaces, CAGD, and Image Processing, held at Biri, Norway. This book contains 48 chapters that include the topics of blossoming, cyclides, data fitting and interpolation, and finding intersections of curves and surfaces. Considerable chapters explore the geometric continuity, geometrical optics, image and signal processing, and modeling of geological structures. The remaining chapters discuss the principles of multiresolution analysis, NURBS, offsets, radial basis functions, rational splines, robotics, spline and Bezier methods for curve and surface modeling, subdivision, terrain modeling, and wavelets. This book will prove useful to mathematicians, computer scientists, and advance mathematics students.

Inhalt

Front Cover 1
Mathematical Methods in Computer Aided Geometric Design II 4
Copyright Page 5
Table of Contents 6
PREFACE 9
PARTICIPANTS 10
Chapter 1. Symmetrizing Multiaffine Polynomials 20
§1. Introduction and Motivation 20
§2. Cubics 22
§3. Quartics, Quintics, and Sextics 23
§4. Observations on Conversion to B-spline Form 25
§5. Open Questions 25
References 26
Chapter 2. Norm Estimates for Inverses of Distance Matrices 28
§1. Introduction 28
§2. The Univariate Case for the Euclidean Norm 29
§3. The Multivariate Case for the Euclidean Norm 32
§4. Fourier Transforms and Bessel Transforms 33
§5. The Least Upper Bound for Subsets of the Integer Grid 36
References 36
Chapter 3. Numerical Treatment of 
38 
§1. Introduction 38
§2. Lattice Evaluation 
40 
§3. Marching Based on Davidenko's Differential Equation 42
§4. Marching Based on Taylor Expansion 45
§5. Conclusion and Future Extensions 46
References 47
Chapter 4. Modeling Closed Surfaces: 
48 
§1. Introduction 48
§2. Subdivision Schemes 49
§3. Discrete Interpolation 51
§4. Algebraic Interpolation 51
§5. Transfinite 
52 
§6. Octree and Face Octree Representations 52
§7. Discussion of These Modeling Schemes 54
References 59
Chapter 5. A New Characterization of Plane 
62 
§1. Introduction 62
§2. A Characterization of Elástica by their Curvature Function 63
§3. A Characterizing Representation Theorem 72
References 75
Chapter 6. POLynomials, POLar Forms, and InterPOLation 76
§1. Introduction 76
§2. Algebraic Definition of Polar Curves 76
§3. Interpolation 81
§4. Conclusion and a Few Historical Remarks 85
Chapter 7. Pyramid Patches Provide 
88 
§1. Introduction 88
§2. Linear Independence of Families of Lineal Polynomials 90
§3. B-patches for Hn(IRs) 97
§4. Other Pyramid Schemes 100
§5. B-patches for IIn 
107 
§6. Degree Raising, Conversion and Subdivision for B-patches 111
References 119
Chapter 8. Implicitizing Rational Surfaces with Base Points by Applying Perturbations and the 
120 
§1. Introduction 120
§2. Mathematical Preliminaries 121
§3. The Factors of Zero Theorem 123
§4. Implicitization with Base Points Using the Dixon Resultant 125
§5. An Implicitization Example 127
§6. Conclusion and Open Problems 128
References 128
Chapter 9. Wavelets and Multiscale Interpolation 130
§1. Introduction 130
§2. Wavelets and Multiresolution 
132 
§3. Fundamental Scaling Functions 136
§4. Symmetric and Compactly Supported Scaling Functions 140
§5. Subdivision Schemes 143
§6. Regularity 146
References 151
Chapter 10. Decomposition of Splines 154
§1. Introduction 154
§2. Decomposition 155
§3. Decomposing Splines 159
§4. Box Spline Decomposition 169
§5. Data Reduction by Decomposition 171
References 178
Chapter 11. A Curve Intersection Algorithm with Processing of Singular Cases: Introduction 
180 
§1. Introduction 180
§2. Clipping 181
§3. Singular Cases 183
§4. Examples 185
§5. Extension to Surfaces 187
§6. Conclusion 188
References 188
Chapter 12. Best Approximations of Parametric 
190 
§1. Introduction 190
§2. Distances in the Set of Plane Parametric Curves 191
§3. Non-linear Approximation Theory for Bézier Curves 196
§4. Numerical Methods and Examples 199
References 202
Chapter 13. An Approximately Cubic Surface Interpolant 204
§1. Introduction 204
§2. Notation and Background 205
§3. Motivation 206
§4. Approximate Continuity 206
§5. Geometric Hermite Curves 207
§6. The Cubic Interpolant 207
§7. A Piecewise Cubic Surface Scheme 210
§8. Results 211
§9. Conclusions 213
§10. Future Work 213
References 214
Chapter 14. On the Continuity of 
216 
§1. Introduction 216
§2. Characterization of G2 Continuity 217
§3. G2 Continuity Between Two Surface Patches 217
§4. G2 Continuity Around an N-patch Corner 218
§5. Piecewise Representation of G2 
225 
References 225
Chapter 15. Stationary and Non-Stationary 
228 
§1. Introduction 228
§2. Analysis of Stationary Schemes 231
§3. Analysis of Non-stationary Schemes 232
References 235
Chapter 16. 
236 
§1. Introduction 236
§2. Properties of General 
237 
§3. Interpolation 239
§4. Local Basis Function 241
§5. Spline Interpolation 243
References 246
Chapter 17. Offset Approximation Improvement 
248 
§1. Introduction 248
§2. Background 249
§3. Getting a Better Approximation of Offsets 250
§4. Examples 252
References 256
Chapter 18. Curves and Surfaces in Geometrical Optics 258
§1. Introduction 258
§2. Envelopes, Offset 
261 
§3. Simple Refraction/reflection 
267 
§4. Concluding Remarks 277
References 278
Chapter 19. Evaluation and Properties of 
280 
§1. Introduction 280
§2. Definitions 283
§3. The Derivative of a Rational B-spline Curve 286
§4. The Hodograph Property 290
§5. Bounds on the Derivative 292
References 293
Chapter 20. Hybrid Cubic Bézier Triangle Patches 294
§1. Introduction 294
§2. Hybrid Cubic Bezier 
295 
§3. Non-parametric Hybrid Patches 298
§4. Cross Boundary Derivatives 300
§5. Concluding Remarks 304
References 304
Chapter 21. Modelling Geological Structures Using Splines 306
§1. Introduction 306
§2. Building Geological Models 307
§3. Program Structures 313
References 314
Chapter 22. Wonderful Triangle: A Simple, Unified, Algorithmic Approach to Change of Basis Procedures in 
316 
§1. Introduction 316
§2. Progressive and Pólya Bases 317
§3. Dual Functionals 321
§4. Wonderful Triangle 322
§5. Algorithms — Knot Insertion 325
§6. Principles of Duality 330
§7. More Algorithms — Knot Deletion 332
§8. Still More Algorithms — Factoring the Transformation 335
§9. Summary, Conclusions, and Future Work 336
References 337
Chapter 23. An Arbitrary Mesh Network Scheme 
340 
§1. Introduction 340
§2. The Mesh Network 341
§3. The Rational Spline Strip Functions 342
§4. Blending the Strip Functions on Polygonal Domains 344
§5. Examples and Concluding Remarks 347
References 348
Chapter 24. Bézier Curves and Surface Patches on Quadrics 350
§1. Introduction 350
§2. A Suitable Map 351
§3. Bezier Curves on Quadrics 352
§4. Bezier Patches on Quadrics 354
§5. General Solution 357
§6. Conclusion 360
References 360
Chapter 25. Monotonicity Preserving Interpolation 
362 
§1. Introduction 362
§2. Rational Cubic Bézier Representation in an Interval 363
§3. The Monotonicity Conditions 364
§4. Determination of the Weights and the First Derivatives 365
§5. Modification of the Curve 366
§6. Numerical Examples 367
§7. Conclusion 369
References 369
Chapter 26. Minimization of Interpolating Spline Curves 
370 
§1. The Problem 370
§2. Minimization on the Unit Interval 371
§3. Examples 373
References 377
Chapter 27. On Piecewise Quadratic 
382 
§1. Introduction 382
§2. Placing the Breakpoints 383
§3. Examples 387
§4. Interpolation 388
§5. More Examples 389
References 389
Chapter 28. Non-affine Blossoms 
390 
§1. Introduction 390
§2. Polynomials of Degree 4 Satisfying a 
392 
§3. Q-spline Curves 396
References 402
Chapter 29. On a Class of Data Parametrizations: 
404 
§1. Introduction 404
§2. Corner Prohibiting Parametrizations 405
§3. Proof of 
407 
§4. Uniform 
411 
References 413
Chapter 30. Wavelets and Image Compression 414
§1. Introduction 414
§2. Wavelet Decomposition of Images: The Haar Transform 415
§3. Image Compression in 
417 
§4. Error, Smoothness, and Quantization 418
§5. Compression in 
. < 8
§6. Examples 423
References 423
Chapter 31. Lower Bounds on the Dimension of Bivariate 
424 
§1. Introduction 424
§2. The Main Result 425
§3. The Lower Bound 426
§4. Generic Triangulations 431
References 434
Chapter 32. Geometrie Contact of Order . 
436 
§1. Introduction 436
§2. Frenet-contact of Order . 436
§3. Comparison Between Fp-contact and 
440 
References 440
Chapter 33. On Non-Parametric Constrained Interpolation 442
§1. Introduction 442
§2. Straight Line as Constraint Curve 444
§3. Quadratic Curve as Constraint Curve 445
§5. Numerical Results 450
§6. Conclusions 452
References 453
Chapter 34. Tensor Product Slices 454
§1. Introduction 454
§2. Tensor Products and Slices 456
§3. Generally Inherited Properties 458
§4. Dimension Raising 458
§5. Ordering of Affine Recurrences 458
§6. Conversion to Bézier Assimilated Form via Blossoming 459
§7. Joining a Slice Patch to an Existing Surface 461
§8. Progress 462
References 463
Chapter 35. Construction of Smooth Surfaces 
464 
§1. Introduction 464
§2. Simple Quadrilateral Meshes 465
§3. Continuity Conditions 465
§4. SQM upon Subdivision of a Cube 466
§5. Further Subdivision of A Cube 470
References 478
Chapter 36. The Virtues of Cyclides in CAGD 480
§1. Surface Representations 480
§2. The Dupin Cyclides 483
§3. Cyclides in Solid Modelling 488
§4. Summary and Conclusions 494
References 494
Chapter 37. Simple Surfaces Have No Simple 
498 
§1. Introduction 498
§2. Simple Closed Surfaces 499
§3. Simple Open Surfaces 500
§4. Concluding Remarks 503
References 503
Chapter 38. Some Tools for Quasi-Interpolation 
504 
Introduction 504
§1. What is a Quasi-interpolant? 505
§2. B-approximation Viewed as a Digital Filter 509
§3. Properties 511
§4. Modified Approximants 513
§5. Examples 516
Conclusion 518
References 519
Chapter 39. Discrete Bézier Curves and Surfaces 520
§1. Introduction 520
§2. Discrete Bernstein Basis 521
§3. Algorithms for Discrete Bezier Curves 522
§4. Subdivision Algorithm 524
§5. Extension Algorithms 527
§6. Joining two Discrete Bézier Curves 528
§7. Rectangular Discrete Bézier Surfaces 532
§s. Triangular Discrete Bezier Surfaces 535
§9. Discrete Bernstein Quasi-interpolants 535
§10. Dual Basis, Condition Number, Finite Differences 536
References 537
Chapter 40. Rational Geometrie Curve Interpolation 540
§1. Introduction 540
§2. Two-point GC2 
543 
§3. Two-point GC1 
548 
§4. Lagrange Interpolation via Hermite Interpolation 550
§5. Direct Lagrange Interpolation in IR3 550
§6. Direct Lagrange Interpolation in IR2 551
§7. Examples 553
References 557
Chapter 41. Curvature Properties of Parametric 
560 
§1. Introduction 560
§2. The Criterion for a Vertex 561
§3. The Global Criterion 564
§4. Classification of Polynomial Degree Two Surfaces 566
References 571
Chapter 42. Offsets of Polynomial Bezier Curves: 
572 
§1. Introduction 572
§2. Interval Bezier Curves 573
§3. Hermite Approximation of Offset Curves 575
§4. Polynomial Approximation of Rational Bezier Curves 577
§5. Numerical Example 579
§6. Discussion 580
References 581
Chapter 43. Representing Piecewise Polynomials as Linear 
582 
§1. Introduction 582
§2. Polynomials and Polar Forms 583
§3. The New B-spline Scheme 585
§4. Piecewise Polynomials as Linear Combinations of B-splines 587
References 589
Chapter 44. An Explicit Derivation of Discretely Shaped 
590 
§1. Introduction 590
§2. Geometric Continuity and Beta-constraints 592
§3. Beta-splines 595
§4. Beta-constraints Revisited 597
§5. Solving for 
600 
§6. Conclusion 605
References 606
Chapter 45. Discrete Convolution Schemes 608
§1. Introduction 608
§2. Discrete Convolution 609
§3. Bernstein Basis as Discrete Convolution 610
§4. Convolution Basis Functions 610
§5. Basic Properties of the Convolution Basis Functions 612
§6. Identities for the Convolution Basis Functions 614
§7. Marsden's Identity and Convolution 614
§8. Relation to the Discrete B-splines 616
§9. Dual Functionals 616
§10. Convolution Curves 616
§11. Summary 618
References 618
Chapter 46. A Method for Removing the Singularities 
620 
§1. Introduction 620
§2. Gregory' s Square 621
§3. Gregory Surfaces 622
§4. Gregory Functions as Rational Bezier Functions 623
§5. Conversion of Gregory Surfaces to Rational Bézier Surfaces 624
§6. Removal of the Singurality of Gregory Functions 627
§7. Conclusion 628
References 628
Chapter 47. Bivariate Spline Approximation 
630 
§1. Introduction 630
§2. Solution of the Problem 631
§3. The Algorithm 633
§4. Numerical Results 634
References 637
Chapter 48. Interpolation with g-splines 638
§1. Introduction 638
§2. The g-Spline Space 639
§3. G1 Constraints 641
§4. A G1 Interpolation Problem and its Solution 642
§5. Interpolating Closed Surfaces 647
References 649

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-5798-3 / 1483257983
• ISBN-13: 978-1-4832-5798-3 / 9781483257983