Introduction to the mathematics of subdivision surfaces:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
2010
|
| Subjects: | |
| Links: | http://www.loc.gov/catdir/enhancements/fy1007/2009047055-b.html http://www.loc.gov/catdir/enhancements/fy1007/2009047055-d.html http://www.loc.gov/catdir/enhancements/fy1007/2009047055-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020349723&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXIV, 356 S. graph. Darst. |
| ISBN: | 9780898716979 |
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| 245 | 1 | 0 | |a Introduction to the mathematics of subdivision surfaces |c Lars-Erik Andersson ; Neil F. Stewart |
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Record in the Search Index
| DE-BY-TUM_call_number | 0048 MAT 652f 2010 B 1557 |
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| adam_text | Titel: Introduction to the mathematics of subdivision surfaces
Autor: Andersson, Lars-Erik
Jahr: 2010
Contents
List of Figures xi
List of Tables xv
Preface xvii
Notation, Conventions, Abbreviations xxi
1 Introduction 1
1.1 A brief overview .......................... 1
1.2 Underlying combinatorial structure................ 9
1.2.1 Polyhedral meshes................... 9
1.2.2 Primal and dual subdivision methods........ 16
1.2.3 Stencils......................... 20
1.3 Examples and classification..................... 21
1.3.1 The methods of Catmull-Clark, Doo-Sabin,
and Loop........................ 22
1.3.2 A classification of subdivision methods ....... 33
1.4 Subdivision matrices ........................ 38
1.4.1 The global subdivision matrix ............ 38
1.4.2 Global subdivision matrices for B-spline
functions........................ 40
1.4.3 Local subdivision matrices .............. 42
1.5 Generating fractal-like objects................... 45
1.6 Additional comments........................ 47
1.7 Exercises............................... 47
1.8 Projects ............................... 49
2 B-Spline Surfaces 51
2.1 Mathematical preliminaries..................... 52
2.2 Univariate uniform B-spline functions............... 55
2.2.1 Definition of B-spline basis functions using
convolution....................... 55
2.2.2 Recursion formulas for uniform B-splines...... 58
VIII
Contents
2.2.3 The Lane-Riesenfeld algorithm............ 67
2.2.4 Two important principles............... 71
2.3 Tensor-product surfaces ...................... 74
2.4 B-spline methods for finite meshes................. 79
2.5 Further results for univariate B-splines.............. 80
2.5.1 Differentiation..................... 80
2.5.2 Partition of unity ................... 83
2.5.3 Linear independence.................. 84
2.5.4 A linear function space................ 85
2.5.5 Computation of exact values............. 86
2.5.6 Application to the tensor-product-surface case ... 90
2.6 Additional comments........................ 91
2.7 Exercises............................... 91
Box-Spline Surfaces 93
3.1 Notation and definitions...................... 93
3.2 Properties of box-spline nodal functions.............. 97
3.3 Continuity properties of box splines................ Ill
3.4 Box-spline subdivision polynomials................ 112
3.5 Centered box-spline subdivision.................. 116
3.5.1 Centered nodal functions and subdivision
polynomials....................... 116
3.5.2 Recursion formulas for control points of
box-spline surfaces................... 117
3.6 Partition of unity and linear independence for box splines .... 126
3.7 Box-spline methods and variants for finite meshes........ 130
3.7.1 The Loop method and its extension to higher
orders.......................... 131
3.7.2 The Midedge and 4-8 subdivision methods ..... 133
3.8 Additional comments........................ 141
3.9 Exercises............................... 141
Generalized-Spline Surfaces 145
4.1 General subdivision polynomials.................. 146
4.2 General-subdivision-polynomial methods and their variants . . . 149
4.2.1 Examples of non-box-spline schemes: 4pt * 4pt,
Butterfly, /3-subdivision (regular case)....... 149
4.2.2 Comparison of 4-8 and /3-subdivision
(regular case)...................... 159
4.2.3 Some Generalized-spline methods: /3-subdivision.
Modified Butterfly, and Kobbelt........... 161
4.3 Fourier analysis of nodal functions................. 166
4.4 Support of nodal functions generated by subdivision
polynomials............................. 170
4.5 Affine invariance for subdivision defined by a subdivision
polynomial.............................. 172
Contents
4.6 A two-dimensional manifold serving as parametric domain . . . 173
4.6.1 The two-dimensional manifold............ 173
4.6.2 A topology on the manifold.............. 176
4.6.3 Local homeomorphisms................ 178
4.6.4 Construction of the local homeomorphisms..... 178
4.7 Generalized splines and Generalized-spline subdivision
methods............................... 181
4.8 Additional comments........................ 186
4.9 Exercises............................... 186
Convergence and Smoothness 189
5.1 Preliminary results for the regular case.............. 190
5.2 Convergence of box-spline subdivision processes......... 197
5.2.1 Linear convergence................... 198
5.2.2 Quadratic convergence................. 202
5.3 Convergence and smoothness for general subdivision
polynomials ............................. 207
5.3.1 Convergence analysis ................. 208
5.3.2 Smoothness....................... 215
5.4 General comments on the nonregular case ............ 219
5.5 Convergence for the nonregular case (example of
Catmull-Clark)........................... 223
5.5.1 The parametric domain................ 223
5.5.2 Spectral analysis.................... 224
5.5.3 Convergence...................... 232
5.6 Smoothness analysis for the Catmull-Clark scheme....... 234
5.7 Conditions for single sheetedness.................. 239
5.8 Further reading on convergence and smoothness......... 243
5.9 Additional comments........................ 245
5.10 Exercises............................... 245
5.11 Project................................ 246
Evaluation and Estimation of Surfaces 247
6.1 Evaluation and tangent stencils for nonregular points...... 248
6.1.1 Evaluation stencils................... 248
6.1.2 Tangent stencils.................... 251
6.2 Evaluation and tangent stencils for subdivision-polynomial
methods............................... 255
6.2.1 Evaluation of nodal functions at grid points..... 256
6.2.2 Evaluation of derivatives of the nodal functions . . . 258
6.3 Exact parametric evaluation.................... 260
6.3.1 A method of de Boor ................. 260
6.3.2 Stam s method..................... 263
6.4 Precision sets and polynomial reproduction............ 266
6.4.1 Conditions for given reproduction and generation
degree.......................... 267
x Contents
6.4.2 Polynomial precision for box splines......... 274
6.4.3 Polynomial precision for non-box-splines....... 276
6.5 Bounding envelopes for patches.................. 280
6.5.1 Modified nodal functions ............... 281
6.5.2 Bounding linear functions............... 282
6.6 Adaptive subdivision........................ 284
6.7 Additional comments........................ 284
6.8 Exercises............................... 284
6.9 Projects ............................... 286
7 Shape Control 287
7.1 Shape control for primal methods................. 287
7.1.1 Surface boundaries and sharp edges......... 288
7.1.2 The Biermann-Levin-Zorin rules for sharp edges . . 292
7.1.3 Interpolation of position and normal direction . . . 294
7.2 Shape control for dual methods.................. 295
7.2.1 Control-point modification.............. 296
7.2.2 Other approaches ................... 297
7.3 Further reading on shape control.................. 297
7.3.1 Subdivision-based multiresolution editing...... 298
7.3.2 Mesh-decimation multiresolution editing....... 303
7.3.3 Other aspects of subdivision-surface shape
control ......................... 304
7.4 Additional comments........................ 304
7.5 Exercises............................... 304
7.6 Projects ............................... 305
Appendix 307
A.l Equivalence of Catmull-Clark rules................ 307
A.2 The complex Fourier transforms and series............ 310
A.2.1 The Fourier transform: Univariate case....... 310
A.2.2 The Fourier transform: Bivariate case........ 312
A.2.3 Complex Fourier series ................ 314
A.2.4 Discrete complex Fourier series............ 314
A.3 Regularity for box-spline nodal functions............. 315
A.3.1 Fourier transforms of box-spline nodal functions . . 315
A.3.2 Proof of Theorem 3.3.2................ 317
A.4 Products of convergent subdivision polynomials......... 320
A.5 Exercises............................... 323
Notes
Bibliography
Index
325
337
349
|
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| language | English |
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| spellingShingle | Andersson, Lars-Erik Introduction to the mathematics of subdivision surfaces Subdivision surfaces (Geometry) Fläche (DE-588)4129864-0 gnd Geometrische Modellierung (DE-588)4156717-1 gnd Unterteilungsalgorithmus (DE-588)4753239-7 gnd Spline (DE-588)4182391-6 gnd |
| subject_GND | (DE-588)4129864-0 (DE-588)4156717-1 (DE-588)4753239-7 (DE-588)4182391-6 |
| title | Introduction to the mathematics of subdivision surfaces |
| title_auth | Introduction to the mathematics of subdivision surfaces |
| title_exact_search | Introduction to the mathematics of subdivision surfaces |
| title_full | Introduction to the mathematics of subdivision surfaces Lars-Erik Andersson ; Neil F. Stewart |
| title_fullStr | Introduction to the mathematics of subdivision surfaces Lars-Erik Andersson ; Neil F. Stewart |
| title_full_unstemmed | Introduction to the mathematics of subdivision surfaces Lars-Erik Andersson ; Neil F. Stewart |
| title_short | Introduction to the mathematics of subdivision surfaces |
| title_sort | introduction to the mathematics of subdivision surfaces |
| topic | Subdivision surfaces (Geometry) Fläche (DE-588)4129864-0 gnd Geometrische Modellierung (DE-588)4156717-1 gnd Unterteilungsalgorithmus (DE-588)4753239-7 gnd Spline (DE-588)4182391-6 gnd |
| topic_facet | Subdivision surfaces (Geometry) Fläche Geometrische Modellierung Unterteilungsalgorithmus Spline |
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