Delaunay Configuration B-Splines:
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| Format: | Abschlussarbeit Buch |
| Sprache: | English |
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Passau
April 2022
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| Online-Zugang: | kostenfrei kostenfrei Inhaltsverzeichnis |
| Beschreibung: | xxiii, 223 Seiten Illustrationen |
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Datensatz im Suchindex
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| adam_text | Contents 1 Introduction 1.1 Motivation................................................................................................. 1.2 1.3 1 1 Contributions........................................................................................... 4 Structure .................................................................................................. 5 2 Univariate Splines 2.1 From Polynomials to Splines................................................................... 7 7 2.1.1 2.1.2 2.1.3 Polynomials.................................................................................. 7 Piecewise Polynomials................................................................ 10 Piecewise Linear Interpolation.................................................... 11 2.1.4 Cubic Spline Interpolation.......................................................... 2.2 Definition of Univariate Splines............................................................ 13 16 2.2.1 The Spline Space......................................................................... 16 Divided Differences...................................................................... B-Splines...................................................................................... 2.3 Properties of Univariate Splines............................................................ 2.3.1 Properties of B-Splines................................................................ 2.3.2 Continuity and Derivatives.......................................................... 17 19 2.2.2 2.2.3 22 22 23 2.3.3 Polynomial Reproduction
.......................................................... 25 2.3.4 Approximation............................................................................ 28 2.3.5 Knot Insertion............................................................................... 30 2.3.6 Condition of the В-Spline Basis................................................ 30 3 Simplex Splines 33 3.1 General Multivariate Concepts............................................................... 33 3.1.1 Simplices and Barycentric Coordinates.................................... 33 3.1.2 Multivariate Polynomials............................................................. 38 3.2 Definition of Simplex Splines ................................................................. 3.2.1 Univariate Geometric Interpretation ....................................... 39 39 3.2.2 Multivariate Generalization....................................................... 42 3.3 Properties of Simplex Splines................................................................. 44 xi
3.4 Examples of Simplex Splines.................................................................... 4 Multivariate Spline Spaces 4.1 48 53 Constructing Multivariate Spline Spaces............................................. 53 4.1.1 Generalizing the В-Spline Basis................................................. 53 4.1.2 The Fundamental Problem.......................................................... 54 DMS-Splines ........................................................................................... 4.2.1 A Geometric Approach................................................................ 56 57 58 4.2.3 A Combinatorial Approach.......................................................... В-Patches and B-Weights............................................................. 4.2.4 В-Weights and Simplex Splines................................................. 64 4.2.5 Definition and Properties of DMS-Splines.................................. 65 4.2.6 DMS-Splines and the Fundamental Problem........................... 4.3 Delaunay Configuration B-Splines......................................................... 69 70 4.2 4.2.2 4.4 4.3.1 4.3.2 Voronoi Diagrams......................................................................... 70 Delaunay Triangulations............................................................. 72 4.3.3 Duality Relations.......................................................................... 74 4.3.4 Higher-Order Voronoi Diagrams................................................. 77 4.3.5 Delaunay
Configurations............................................................. 79 4.3.6 Elimination of Duplicates .......................................................... 84 4.3.7 DCB-Splines and the Fundamental Problem........................... 88 4.3.8 Related Work................................................................................. 91 Other Approaches..................................................................................... 92 4.4.1 Tensor Product Splines................................................................. 92 4.4.2 Polyhedral Splines and Box Splines............................................ 94 4.4.3 Generalized Delaunay Configurations......................................... 94 4.4.4 Splines on Triangulations........................................................... 95 5 Local Finiteness 5.1 5.2 62 97 Preliminaries........................................................................................... 97 5.1.1 Problem Statement........................................................................ 97 5.1.2 Prerequisites ................................................................................. 99 Dens^y Cones............................................................................................. 107 5.2.1 Definition and Properties of Cones.............................................. 108 5.2.2 Density Description Using Cones ................................................. Ill 5.2.3 Uniformity of Density Cones.......................................................... 113 5.3 Geometry of Conesand
Balls....................................................................... 117 5.3.1 Geometry of Balls............................................................................. 117 5.3.2 Interactionsof Cones and Balls..................................................... 123 xii
5.4 Local Finiteness Theorem ......................................................................... 129 5.4.1 Generalization to Compact Sets....................................................129 5.4.2 Main Result..................................................................................... 132 5.5 Consequences.............................................................................................. 134 5.5.1 Corollaries.........................................................................................134 5.5.2 Local Finiteness Condition............................................................. 136 6 Knot Insertion 6.1 6.2 143 Preliminaries.............................................................................................. 143 Necessary Criterion..................................................................................... 146 6.2.1 Limits of Differentiability ............................................................. 146 6.2.2 Definition of the Criterion............................................................. 150 6.3 Sufficiency of the Criterion......................................................................... 153 6.3.1 Knot Insertion Formula................................................................... 154 6.3.2 Critical Configurations................................................................... 156 6.3.3 Companion Configurations............................................................. 169 7 Approximation and Condition 195 7.1 Approximation
...........................................................................................195 7.1.1 Multivariate Schoenberg Operator ............................................. 196 7.1.2 Approximation Quality.................................................................. 197 7.2 Condition.................................................................................................... 201 7.2.1 Bounding the Spline Value............................................................ 201 7.2.2 8 Conclusion Bounding the Coefficients............................................................ 202 207 8.1 Summary..................................................................................................... 207 8.2 Practical Considerations............................................................................ 208 8.2.1 Efficient Evaluation of Simplex Splines....................................... 208 8.2.2 Degenerate and Finite Knot Sets ................................................ 210 8.2.3 Computing Delaunay Configurations.......................................... 211 8.3 Future Research Topics............................................................................... 213 Bibliography 217 ί xiii
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| adam_txt |
Contents 1 Introduction 1.1 Motivation. 1.2 1.3 1 1 Contributions. 4 Structure . 5 2 Univariate Splines 2.1 From Polynomials to Splines. 7 7 2.1.1 2.1.2 2.1.3 Polynomials. 7 Piecewise Polynomials. 10 Piecewise Linear Interpolation. 11 2.1.4 Cubic Spline Interpolation. 2.2 Definition of Univariate Splines. 13 16 2.2.1 The Spline Space. 16 Divided Differences. B-Splines. 2.3 Properties of Univariate Splines. 2.3.1 Properties of B-Splines. 2.3.2 Continuity and Derivatives. 17 19 2.2.2 2.2.3 22 22 23 2.3.3 Polynomial Reproduction
. 25 2.3.4 Approximation. 28 2.3.5 Knot Insertion. 30 2.3.6 Condition of the В-Spline Basis. 30 3 Simplex Splines 33 3.1 General Multivariate Concepts. 33 3.1.1 Simplices and Barycentric Coordinates. 33 3.1.2 Multivariate Polynomials. 38 3.2 Definition of Simplex Splines . 3.2.1 Univariate Geometric Interpretation . 39 39 3.2.2 Multivariate Generalization. 42 3.3 Properties of Simplex Splines. 44 xi
3.4 Examples of Simplex Splines. 4 Multivariate Spline Spaces 4.1 48 53 Constructing Multivariate Spline Spaces. 53 4.1.1 Generalizing the В-Spline Basis. 53 4.1.2 The Fundamental Problem. 54 DMS-Splines . 4.2.1 A Geometric Approach. 56 57 58 4.2.3 A Combinatorial Approach. В-Patches and B-Weights. 4.2.4 В-Weights and Simplex Splines. 64 4.2.5 Definition and Properties of DMS-Splines. 65 4.2.6 DMS-Splines and the Fundamental Problem. 4.3 Delaunay Configuration B-Splines. 69 70 4.2 4.2.2 4.4 4.3.1 4.3.2 Voronoi Diagrams. 70 Delaunay Triangulations. 72 4.3.3 Duality Relations. 74 4.3.4 Higher-Order Voronoi Diagrams. 77 4.3.5 Delaunay
Configurations. 79 4.3.6 Elimination of Duplicates . 84 4.3.7 DCB-Splines and the Fundamental Problem. 88 4.3.8 Related Work. 91 Other Approaches. 92 4.4.1 Tensor Product Splines. 92 4.4.2 Polyhedral Splines and Box Splines. 94 4.4.3 Generalized Delaunay Configurations. 94 4.4.4 Splines on Triangulations. 95 5 Local Finiteness 5.1 5.2 62 97 Preliminaries. 97 5.1.1 Problem Statement. 97 5.1.2 Prerequisites . 99 Dens^y Cones. 107 5.2.1 Definition and Properties of Cones. 108 5.2.2 Density Description Using Cones . Ill 5.2.3 Uniformity of Density Cones. 113 5.3 Geometry of Conesand
Balls. 117 5.3.1 Geometry of Balls. 117 5.3.2 Interactionsof Cones and Balls. 123 xii
5.4 Local Finiteness Theorem . 129 5.4.1 Generalization to Compact Sets.129 5.4.2 Main Result. 132 5.5 Consequences. 134 5.5.1 Corollaries.134 5.5.2 Local Finiteness Condition. 136 6 Knot Insertion 6.1 6.2 143 Preliminaries. 143 Necessary Criterion. 146 6.2.1 Limits of Differentiability . 146 6.2.2 Definition of the Criterion. 150 6.3 Sufficiency of the Criterion. 153 6.3.1 Knot Insertion Formula. 154 6.3.2 Critical Configurations. 156 6.3.3 Companion Configurations. 169 7 Approximation and Condition 195 7.1 Approximation
.195 7.1.1 Multivariate Schoenberg Operator . 196 7.1.2 Approximation Quality. 197 7.2 Condition. 201 7.2.1 Bounding the Spline Value. 201 7.2.2 8 Conclusion Bounding the Coefficients. 202 207 8.1 Summary. 207 8.2 Practical Considerations. 208 8.2.1 Efficient Evaluation of Simplex Splines. 208 8.2.2 Degenerate and Finite Knot Sets . 210 8.2.3 Computing Delaunay Configurations. 211 8.3 Future Research Topics. 213 Bibliography 217 ί xiii |
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| spelling | Schlenker, Florian Verfasser aut Delaunay Configuration B-Splines Florian Schenker Passau April 2022 xxiii, 223 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Dissertation Universität Passau 2022 Spline (DE-588)4182391-6 gnd rswk-swf Delaunay-Triangulierung (DE-588)4628829-6 gnd rswk-swf Spline-Raum (DE-588)4626863-7 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Delaunay-Triangulierung (DE-588)4628829-6 s Spline-Raum (DE-588)4626863-7 s Spline (DE-588)4182391-6 s DE-604 Erscheint auch als Online-Ausgabe urn:nbn:de:bvb:739-opus4-11225 https://opus4.kobv.de/opus4-uni-passau/frontdoor/index/index/docId/1122 Verlag kostenfrei Volltext https://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-11225 Resolving-System kostenfrei Volltext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033790916&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schlenker, Florian Delaunay Configuration B-Splines Spline (DE-588)4182391-6 gnd Delaunay-Triangulierung (DE-588)4628829-6 gnd Spline-Raum (DE-588)4626863-7 gnd |
| subject_GND | (DE-588)4182391-6 (DE-588)4628829-6 (DE-588)4626863-7 (DE-588)4113937-9 |
| title | Delaunay Configuration B-Splines |
| title_auth | Delaunay Configuration B-Splines |
| title_exact_search | Delaunay Configuration B-Splines |
| title_exact_search_txtP | Delaunay Configuration B-Splines |
| title_full | Delaunay Configuration B-Splines Florian Schenker |
| title_fullStr | Delaunay Configuration B-Splines Florian Schenker |
| title_full_unstemmed | Delaunay Configuration B-Splines Florian Schenker |
| title_short | Delaunay Configuration B-Splines |
| title_sort | delaunay configuration b splines |
| topic | Spline (DE-588)4182391-6 gnd Delaunay-Triangulierung (DE-588)4628829-6 gnd Spline-Raum (DE-588)4626863-7 gnd |
| topic_facet | Spline Delaunay-Triangulierung Spline-Raum Hochschulschrift |
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