Mathematical Methods in Computer Aided Geometric Design (eBook)

Tom Lyche, Larry L. Schumaker (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
628 Seiten
Elsevier Science (Verlag)
978-1-4832-5780-8 (ISBN)

Infos

Mathematical Methods in Computer Aided Geometric Design covers the proceedings of the 1988 International Conference by the same title, held at the University of Oslo, Norway. This text contains papers based on the survey lectures, along with 33 full-length research papers. This book is composed of 39 chapters and begins with surveys of scattered data interpolation, spline elastic manifolds, geometry processing, the properties of B‚zier curves, and Gr”bner basis methods for multivariate splines. The next chapters deal with the principles of box splines, smooth piecewise quadric surfaces, some applications of hierarchical segmentations of algebraic curves, nonlinear parameters of splines, and algebraic aspects of geometric continuity. These topics are followed by discussions of shape preserving representations, box-spline surfaces, subdivision algorithm parallelization, interpolation systems, and the finite element method. Other chapters explore the concept and applications of uniform bivariate hermite interpolation, an algorithm for smooth interpolation, and the three B-spline constructions. The concluding chapters consider the three B-spline constructions, design tools for shaping spline models, approximation of surfaces constrained by a differential equation, and a general subdivision theorem for B‚zier triangles. This book will prove useful to mathematicians and advance mathematics students.

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Mathematical Methods in Computer Aided Geometric Design covers the proceedings of the 1988 International Conference by the same title, held at the University of Oslo, Norway. This text contains papers based on the survey lectures, along with 33 full-length research papers. This book is composed of 39 chapters and begins with surveys of scattered data interpolation, spline elastic manifolds, geometry processing, the properties of Bezier curves, and Grobner basis methods for multivariate splines. The next chapters deal with the principles of box splines, smooth piecewise quadric surfaces, some applications of hierarchical segmentations of algebraic curves, nonlinear parameters of splines, and algebraic aspects of geometric continuity. These topics are followed by discussions of shape preserving representations, box-spline surfaces, subdivision algorithm parallelization, interpolation systems, and the finite element method. Other chapters explore the concept and applications of uniform bivariate hermite interpolation, an algorithm for smooth interpolation, and the three B-spline constructions. The concluding chapters consider the three B-spline constructions, design tools for shaping spline models, approximation of surfaces constrained by a differential equation, and a general subdivision theorem for Bezier triangles. This book will prove useful to mathematicians and advance mathematics students.

Inhalt

Front Cover 1
Mathematical Methods in Computer Aided Geometric Design 4
Copyright Page 5
Table of Contents 6
PREFACE 9
PARTICIPANTS 10
Chapter 1. Scattered Data Interpolation in Three or More Variables 18
§1. Introduction 18
§2. Rendering of Trivariate Functions 24
§3. Tensor Product Schemes 25
§4. Point Schemes 26
§5. Natural Neighbor Interpolation 29
§6. k-dimensional Triangulations 30
§7. Tetrahedral Schemes 32
§8. Simplicial Schemes 36
§9. Multivariate Splines 39
§10. Transfinite Hypercubal Methods 42
§11. Derivative Generation 42
§12. Interpolation on the sphere and other surfaces 44
§13. Conclusions 44
Acknowledgments 46
References 46
Chapter 2. Some Applications of Discrete Dm Splines 52
§1. Vh-Discrete Smoothing Dm-Splines 52
§3. Approximation of non-Regular Functions 55
References 60
Chapter 3. Spline Elastic Manifolds 62
§1. Introduction 62
§2. The Space k 63
§3. The Hilbert Kernel of k 65
§4. Characterization of the elastic spline manifold 67
References 67
Chapter 4. Geometry Processing: Curvature Analysis and Surface-Surface Intersection 68
§1. Introduction to Geometry Processing 68
§2. Curve Fairing 69
§3. Surface Curvature Analysis 69
§4. Surface-Surface Intersection 72
§5. Offset Surfaces 74
Acknowledgments 76
References 76
Chapter 5. Three Examples of Dual Properties of Bézier Curves 78
§1. Introduction 78
§2. Example 1. Degree Elevation and Differentiation 79
§3. Example 2. Transformations to and from Monomial Form 81
§4. Example 3. de Casteljau and Horner Evaluation 82
§5. Concluding Remarks 86
References 86
Chapter 6. What is the Natural Generalization of a Bézier Curve? 88
§1. Introduction 88
§2. The Canonical Split 89
§3. B-Spline Properties 92
§4. Pólya Properties 94
§5. Shared Properties and Dual Properties 98
§6. Conclusions 100
References 101
Chapter 7. Convexity and a Multidimensional Version of the Variation Diminishing Property of Bernstein Polynomials 104
§1. Notation and Definitions 104
§2. Piecewise Linear Surface Over a Convex Polyhedron 105
§3. Variation Diminishing Property of Bernstein Polynomials 107
Acknowledgment 109
References 109
Chapter 8. Gröbner Basis Methods for Multivariate Splines 110
§1. Dimensions of Spline Spaces 110
§2. Gröbner Bases 113
§3. Computing Dimension Series and Bases of Splines 116
§4. Example and Discussion 117
References 119
Chapter 9. On Finite Element Interpolation Problems 122
§1. Introduction 122
§2. Interpolation Systems in IR 122
§3. Interpolation Problem Associated to an Interpolation System 123
§4. Argyris Triangle 124
§5. Construction of the Solution of the Interpolation Problem 126
§6. Basic Functions for the Argyris Triangle 128
References 129
Chapter 10. The Design of Curves and Surfaces by Subdivision Algorithms 132
§1. Introduction 132
§2. The Algorithms of de Casteljau and Chaikin 133
§3. De Rham's Construction of Certain Planar Curves 138
§4. Algorithms for Surfaces 140
§5. The Subdivision Algorithm for Bernstein-Bézier Curves 141
§6. Subdivision Algorithms for Univariate Spline Functions 144
§7. Cube Splines and the Line Average Algorithm 148
§8. Rates of Convergence 154
§9. Subdivision as Corner Cutting 156
§10. A Matrix Approach to Subdivision for the Univariate Case 158
§11. Regular Subdivision 162
References 167
Chapter 11. A Data Dependent Parametrization for Spline Approximation 172
§1. Introduction 172
§2. Background 174
§3. Finding an Initial Parametrization 176
§4. Experimental Results 180
§5. Remarks 182
References 182
Chapter 12. On the Evaluation of Box Splines 184
§1. Introduction 184
§2. Boxes, Cones and Simplex Splines 185
§3. Regular Meshes 186
§4. The Bivariate Case 187
§5. The Trivariate Case 191
Acknowledgment 195
References 195
Chapter 13. Smooth Piecewise Quadric Surfaces 198
§1. Introduction 198
§2. Topology of Interpolating Surfaces and Transversal Systems 199
§3. Implicit Bézier Patches 201
§4. Macro Patches for Piecewise Quadric Surfaces 202
References 210
CHapter 14. Inserting New Knots Into Beta-Spline Curves 212
§1. Introduction 212
§2. Generalized Cubic Splines and ß-splines 214
§3. Knot Insertion 216
§4. Applications in CAGD 218
References 221
Chapter 15. Recursive Subdivision and Iteration in Intersections and Related Problems 224
§1. Introduction 224
§2. Comparison of Iteration and Recursive Subdivision 225
§3. General Layout of the Intersection Strategy 227
§4. Critical Subalgorithms 229
§5. Conclusion 231
References 231
Chapter 16. Rational Curves and Surfaces 232
§1. Introduction 232
§2. Conies as Rational Quadratics 232
§3. Derivatives 235
§4. Classification 236
§6. Rational Bezier Curves 238
§7. The de Casteljau Algorithm 239
§8. Derivatives 241
§9. Reparametrization and Degree Elevation 242
§10. Functional Rational Bézier Curves 244
§11. Rational Cubic B-spline Curves 245
§12. Rational B-splines of Arbitrary Degree 247
§13. Reparametrizations of Rational B-spline Curves 248
§14. Interpolation with Rational Cubics 248
§15. Rational B-spline Surfaces 250
Chapter 17. Algebraic Aspects of Geometric Continuity 334
§1. Introduction 334
§2. Some Algebraic Aspects of Geometric Continuity 335
§4. The Invariance of Frenet Frame Continuity Under Projection 342
§5. Some Further Algebraic Aspects of Geometric Continuity 347
§6. Conclusion 352
References 352
Chapter 18. Shape Preserving Representations 354
§1. Introduction 354
§2. Total Positivity 356
§2. Total Positivity 356
§3. Consequences of Total Positivity 359
§4. Other Properties 363
§5. Surfaces 365
References 369
Chapter 19. Geometric Continuity 374
§1. Introduction 374
§2. Geometric Arc Length Continuity for Curves 375
§3. Differential Geometry of Curves 378
§4. Geometric Frenet Frame Continuity for Curves 380
§5. Geometric Continuity for Surfaces 382
§6. Applications 386
References 388
Chapter 20. Curvature Continuous Triangular Interpolants 394
§1. Introduction 394
§2. Preliminary Remarks and Statement of the Problem 394
§3. The Geometric Hermite-operator 395
§4. A Transfinite VC2-patch 396
§5. Discretization of the Transfinite VC2-patch 400
§6. Curvature Estimation for Smooth Surface Design 404
References 405
Chapter 21. Box-Spline Surfaces 406
§1. Introduction 406
§2. Definition and Basic Properties 407
§3. Cardinal Splines 413
§4. Subdivision Algorithms 415
§5. Selected Theorems 420
§6. Bibliographic Comments 422
References 422
Chapter 22. Parallelization of the Subdivision Algorithm for Intersection of Bézier Curves on the FPS T20 424
§1. Introduction 424
§2. The FPS T20 424
3. Parallelization of an Algorithm 425
§4. The Bézier Algorithm 425
§5. Parallelization 426
§6. Vectorization 428
§7. Experimental Results 429
§8. Conclusion 431
References 432
Chapter 23. Composite Quadrilateral Finite Elements of Class C 434
§1. Introduction 434
§2. FVS Quadrilaterals of Class C 435
§3. FVS Quadrilaterals of Class C 437
References 438
Chapter 4. A Knot Removal Strategy for Scattered Data in R 440
§1. Introduction 440
§2. Measure of the Significance of Each Point 441
§4. Control Points From One Triangle to Another 443
§5. On Bell's Approximant 444
§6. Knot Removal 445
§7. Remarks and Comments 446
References 446
Chapter 24. Interpolation Systems and the Finite Element Method 448
§1. Introduction 448
§2. Notation, Definitions, and Previous Results 449
§3. Triangular Finite Elements of Class k and Type 450
§4. Remarks 453
5. Ck-continuity of Finite Elements of Type 454
References 454
Chapter 25. Uniform Bivariate Hermite Interpolation 456
§1. Introduction 456
§2. Interpolations with Few Knots 458
§3. Uniform Hermite Interpolation with Second Derivatives 459
§4. The General Case 464
References 465
Chapter 26. A Survey of Applications of an Affine Invariant Norm 466
§1. Introduction 466
§2. Applications to Scattered Data Interpolation 470
§3. Applications to Knot Selection 476
§4. Applications to Triangulations 478
Acknowledgments 484
References 487
Chapter 27. An Algorithm for Smooth Interpolation to Scattered Data in R 490
§1. Notation 490
§2. Interpolation Problem 491
§3. Choice of the Degree k 492
§4. A Representation for Elements of the Space Sm2m+1 494
§5. Outline of the Algorithm 497
§6. Numerical Experiments 499
References 500
Chapter 28. Some Remarks on Three B-Spline Constructions 502
§1. Introduction 502
§2. Bézier Polynomials 502
§3. de Boor's Algorithm 503
§4. Boehm's Construction 504
§5. A New Development of B-splines 504
§6. Proving de Boor's Algorithm 506
References 507
Chapter 29. Modified B-Spline Approximation for Quasi-Interpolation or Filtering 510
§1. Introduction 510
§2. Transfer Function 511
§3. Choosing the Coefficients bj 513
§4. Conclusions 519
References 519
Chapter 30. Design Tools for Shaping Spline Models 520
§1. Introduction 520
§2. Surfaces from Curves 521
§3. Making Solids from Surfaces 533
§4. Modifying Surface Shapes 535
§5. Examples 536
§6. Conclusions 537
Acknowledgements 538
References 538
Chapter 31. A Process Oriented Design Method for Three-dimensional CAD Systems 542
§1. Introduction 542
§2. Process Oriented Solid Modeling 543
§3. Modeling of Complex Shaped Bodies 545
§4. Examples 547
§5. Conclusion 549
References 549
Chapter 32. Open Questions in the Application of Multivariate B-splines 550
§1. Introduction 550
§2. Multivariate B-splines 550
§3. Scattered Data Contouring 551
§4. Field Analysis 555
§5. Multivariate B-splines Still Outside the Multivariate Theory 556
References 557
Chapter 33. On Global GC2 Convexity Preserving Interpolation of Planar Curves by Piecewise Bézier Polynomials 560
§1. Global Parametric Spline Interpolation 560
§2. Cubic Pieces at Inflection Points 562
§3. Straight Sections 565
§4. Not-a-knot Boundary Conditions 566
References 568
Chapter 34. Best Interpolation with Free Nodes by Closed Curves 570
§1. Introduction 570
§2. Existence 571
§3. Uniqueness 575
§4. Discussion 579
References 580
Chapter 35. Segmentation Operators on Coons' Patches 582
§1. Introduction 582
§2. Introduction to the Theory of Coons' Patches 583
§3. Segmentation Operators on Coons' Patches 587
§4. Segmentation of Subclasses 589
§5. Segmentation of Tensor-Product Surfaces 591
References 593
Chapter 36. neral Subdivision Theorem for Bézier Triangles 594
§1. Introduction 594
§2. The Blossoming Principle 595
§3. Subdivision Algorithms for Bézier Triangles 597
§4. Conclusions 601
References 601
Chapter 37. Cardinal Interpolation with Translates of Shifted Bivariate Box-Splines 604
§1. Introduction 604
§2. Exponential Euler Splines 605
§3. Correctness of Cardinal Interpolation: General Results 606
§4. Correctness of Cardinal Interpolation: Low Order Splines 607
§5. Further Remarks 612
Acknowledgement 612
References 612
Chapter 38. Approximation of Surfaces Constrained by a Differential Equation Using Simplex Splines 614
§1. Introduction 614
§2. Bivariate Simplex Splines 615
§3. Surfaces Constrained by a Differential Equation 617
§4. Inner Products of Lowest Order Splines 618
§5. Evaluation of a Model Problem 618
References 620
Chapter 39. A Construction for VC1 Continuity of Rational Bézier Patches 622
§1. Introduction 622
§2. Tangential Directions for Rectangular Patches 623
§3. Tangential Directions for Triangular Patches 624
§4. The VC1 Construction 626
§5. Farin's Transition Principle 630
§6. Conclusions 631
References 631

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-5780-0 / 1483257800
• ISBN-13: 978-1-4832-5780-8 / 9781483257808