Generalized curvatures:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2008
|
| Schriftenreihe: | Geometry and computing
2 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 261 - 264 |
| Beschreibung: | XI, 266 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9783540737919 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV023425639 | ||
| 003 | DE-604 | ||
| 005 | 20130327 | ||
| 007 | t | ||
| 008 | 080731s2008 gw ad|| |||| 00||| eng d | ||
| 015 | |a 08,A28,1151 |2 dnb | ||
| 016 | 7 | |a 989188892 |2 DE-101 | |
| 020 | |a 9783540737919 |c Pp. : EUR 85.55 (freier Pr.), sfr 139.50 (freier Pr.) |9 978-3-540-73791-9 | ||
| 020 | |z 9780540737919 |c Pp. : EUR 85.55 (freier Pr.), sfr 139.50 (freier Pr.) |9 978-0-540-73791-9 | ||
| 024 | 3 | |a 9783540737919 | |
| 028 | 5 | 2 | |a 12096879 |
| 035 | |a (OCoLC)166372591 | ||
| 035 | |a (DE-599)DNB989188892 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 0 | |a eng | |
| 044 | |a gw |c XA-DE-BE | ||
| 049 | |a DE-824 |a DE-703 |a DE-20 |a DE-188 |a DE-83 | ||
| 050 | 0 | |a QA639.5 | |
| 082 | 0 | |a 516.3/62 |2 22 | |
| 082 | 0 | |a 516.362 |2 22/ger | |
| 084 | |a SK 370 |0 (DE-625)143234: |2 rvk | ||
| 084 | |a 52B10 |2 msc | ||
| 084 | |a 510 |2 sdnb | ||
| 084 | |a 53C20 |2 msc | ||
| 084 | |a 53A04 |2 msc | ||
| 084 | |a 52C10 |2 msc | ||
| 100 | 1 | |a Morvan, Jean-Marie |e Verfasser |0 (DE-588)135744229 |4 aut | |
| 245 | 1 | 0 | |a Generalized curvatures |c Jean-Marie Morvan |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a XI, 266 S. |b Ill., graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Geometry and computing |v 2 | |
| 500 | |a Literaturverz. S. 261 - 264 | ||
| 650 | 4 | |a Convex geometry | |
| 650 | 4 | |a Curvature | |
| 650 | 4 | |a Discrete geometry | |
| 650 | 0 | 7 | |a Krümmungsmaß |0 (DE-588)4242871-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Krümmungsmaß |0 (DE-588)4242871-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-540-73792-6 |
| 830 | 0 | |a Geometry and computing |v 2 |w (DE-604)BV022959018 |9 2 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016608001&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016608001 | ||
Datensatz im Suchindex
| _version_ | 1804137820396191744 |
|---|---|
| adam_text | Contents
1 Introduction................................................... 1
1.1 Two Fundamental Properties................................. 1
1.2 Different Possible Classifications............................. 2
1.3 Part I: Motivation .......................................... 3
1.4 Part II: Background - Metric and Measures..................... 4
1.5 Part III: Background - Polyhedra and Convex Subsets ............ 4
1.6 Part IV: Background - Classical Tools on Differential Geometry ... 5
1.7 Part V: On Volume......................................... 6
1.8 Part VI: The Steiner Formula................................. 6
1.9 Part VII: The Theory of Normal Cycles........................ 7
1.10 Part VIII: Applications to Curves and Surfaces.................. 9
Parti Motivations
2 Motivation: Curves............................................. 13
2.1 The Length of a Curve...................................... 13
2.1.1 The Length of a Segment and a Polygon................. 13
2.1.2 The General Definition ............................... 14
2.1.3 The Length of a C1-Curve.....................:....... 15
2.1.4 An Obvious Convergence Result....................... 16
2.1.5 Warning! Negative Results............................ 16
2.2 The Curvature of a Curve.................................... 17
2.2.1 The Pointwise Curvature of a Curve .................... 17
2.2.2 The Global (or Total) Curvature........................ 19
2.3 The Gauss Map of a Curve................................... 21
2.4 Curves in E2............................................... 22
2.4.1 A Pointwise Convergence Result for Plane Curves ........ 22
2.4.2 Warning! A Negative Result on the Approximation
by Conies........................................... 22
2.4.3 The Signed Curvature of a Smooth Plane Curve........... 24
vi Contents
2.4.4 The Signed Curvature of a Plane Polygon................ 26
2.4.5 Signed Curvature and Topology........................ 27
2.5 Conclusion................................................ 28
3 Motivation: Surfaces........................................... 29
3.1 The Area of a Surface....................................... 29
3.1.1 The Area of a Piecewise Linear Surface................. 29
3.1.2 The Area of a Smooth Surface......................... 29
3.1.3 Warning! The Lantern of Schwarz...................... 30
3.2 The Pointwise Gauss Curvature............................... 33
3.2.1 Background on the Curvatures of Surfaces............... 33
3.2.2 Gauss Curvature and Geodesic Triangles ................ 34
3.2.3 The Angular Defect of a Vertex of a Polyhedron.......... 36
3.2.4 Warning! A Negative Result........................... 37
3.2.5 Warning! The Pointwise Gauss Curvature of a Closed
Surface............................................. 39
3.2.6 Warning! A Negative Result Concerning
the Approximation by Quadrics........................ 40
3.3 The Gauss Map of a Surface................................. 41
3.3.1 The Gauss Map of a Smooth Surface.................... 41
3.3.2 The Gauss Map of a Polyhedron........................ 42
3.4 The Global Gauss Curvature................................. 43
3.5 The Volume............................................... 44
Part II Background: Metrics and Measures
4 Distance and Projection......................................... 47
4.1 The Distance Function...................................... 47
4.2 The Projection Map......................................... 49
4.3 The Reach of a Subset...................................... 52
4.4 The Voronoi Diagrams...................................... 55
4.5 The Medial Axis of a Subset................................. 55
5 Elements of Measure Theory.................................... 57
5.1 Outer Measures and Measures................................ 57
5.1.1 Outer Measures...................................... 57
5.1.2 Measures........................................... 58
5.1.3 Outer Measures vs. Measures.......................... 58
5.1.4 Signed Measures..................................... 59
5.1.5 Borel Measures...................................... 60
5.2 Measurable Functions and Their Integrals...................... 60
5.2.1 Measurable Functions................................ 60
5.2.2 Integral of Measurable Functions....................... 61
Contents vii
5.3 The Standard Lebesgue Measure on E^........................ 62
5.3.1 Lebesgue Outer Measure on R and E^.................. 63
5.3.2 Lebesgue Measure on R and E^........................ 64
5.3.3 Change of Variable................................... 64
5.4 Hausdorff Measures ........................................ 65
5.5 Area and Coarea Formula.................................... 66
5.6 Radon Measures........................................... 67
5.7 Convergence of Measures.................................... 67
Part III Background: Polyhedra and Convex Subsets
6 Polyhedra..................................................... 71
6.1 Definitions and Properties of Polyhedra........................ 71
6.2 Euler Characteristic......................................... 74
6.3 Gauss Curvature of a Polyhedron............................. 75
7 Convex Subsets................................................ 77
7.1 Convex Subsets............................................ 77
7.1.1 Definition and Basic Properties......................... 77
7.1.2 The Support Function ................................ 79
7.1.3 The Volume of Convex Bodies......................... 80
7.2 Differential Properties of the Boundary........................ 81
7.3 The Volume of the Boundary of a Convex Body................. 82
7.4 The Transversal Integral and the Hadwiger Theorem............. 84
7.4.1 Notion of Valuation.................................. 84
7.4.2 Transversal Integral.................................. 85
7.4.3 The Hadwiger Theorem............................... 86
Part IV Background: Classical Tools in Differential Geometry
8 Differential Forms and Densities on E^........................... 91
8.1 Differential Forms and Their Integrals......................... 91
8.1.1 Differential Forms on E^ ............................. 91
8.1.2 Integration of ^-Differential Forms on E^............... 93
8.2 Densities.................................................. 94
8.2.1 Notion of Density on E^.............................. 94
8.2.2 Integration of Densities on EN and the Associated Measure . 95
9 Measures on Manifolds......................................... 97
9.1 Integration of Differential Forms.............................. 97
9.2 Density and Measure on a Manifold........................... 98
9.3 The Fubini Theorem on a Fiber Bundle........................ 99
viii Contents
10 Background on Riemannian Geometry...........................101
10.1 Riemannian Metric and Levi-Civita Connexion..................101
10.2 Properties of the Curvature Tensor............................102
10.3 Connexion Forms and Curvature Forms........................103
10.4 The Volume Form..........................................103
10.5 The Gauss-Bonnet Theorem.................................104
10.6 Spheres and Balls..........................................104
10.7 The Grassmann Manifolds...................................105
10.7.1 The Grassmann Manifold G°(N,k) .....................105
10.7.2 The Grassmann Manifold G(N,k)......................106
10.7.3 The Grassmann Manifolds AG(N,k) and AG°(N,k) .......107
11 Riemannian Submanifolds......................................109
11.1 Some Generalities on (Smooth) Submanifolds ..................109
11.2 The Volume of a Submanifold................................112
11.3 Hypersurfaces in E^........................................113
11.3.1 The Second Fundamental Form of a Hypersurface.........113
11.3.2 ¿ -Mean Curvature of a Hypersurface...................114
11.4 Submanifolds in EN of Any Codimension......................115
11.4.1 The Second Fundamental Form of a Submanifold.........115
11.4.2 £ A-Mean Curvatures in Large Codimension..............116
11.4.3 The Normal Connexion...............................116
11.4.4 The Gauss-Codazzi-Ricci Equations....................117
11.5 The Gauss Map of a Submanifold.............................118
11.5.1 The Gauss Map of a Hypersurface......................118
11.5.2 The Gauss Map of a Submanifold of Any Codimension----118
12 Currents......................................................121
12.1 Basic Definitions and Properties on Currents....................121
12.2 Rectifiable Currents.........................................122
12.3 Three Theorems............................................124
PartV On Volume
13 Approximation of the Volume...................................129
13.1 The General Framework.....................................129
13.2 A General Evaluation Theorem for the Volume..................131
13.2.1 Statement of the Main Result..........................131
13.2.2 Proof of Theorem 38.................................131
13.3 An Approximation Result ...................................133
13.4 A Convergence Theorem for the Volume.......................135
13.4.1 The Framework......................................135
13.4.2 Statement of the Theorem.............................137
Contents ix
14 Approximation of the Length of Curves...........................139
14.1 A General Approximation Result.............................139
14.2 An Approximation by a Polygonal Line........................140
15 Approximation of the Area of Surfaces...........................143
15.1 A General Approximation of the Area.........................143
15.2 Triangulations.............................................144
15.2.1 Geometric Invariant Associated to a Triangle.............144
15.2.2 Geometric Invariant Associated to a Triangulation.........145
15.3 Relative Height of a Triangulation Inscribed in a Surface .........145
15.4 A Bound on the Deviation Angle .............................146
15.4.1 Statement of the Result and Its Consequences............146
15.4.2 Proof of Theorem 45.................................147
15.5 Approximation of the Area of a Smooth Surface by the Area
of a Triangulation..........................................150
Part VI The Steiner Formula
16 The Steiner Formula for Convex Subsets..........................153
16.1 The Steiner Formula for Convex Bodies (1840) .................153
16.2 Examples: Segments, Discs, and Balls.........................155
16.3 Convex Bodies in E^ Whose Boundary is a Polyhedron..........158
16.4 Convex Bodies with Smooth Boundary........................159
16.5 Evaluation of the Quermassintegrale by Means of Transversal
Integrals..................................................161
16.6 Continuity of the äE *........................................162
16.7 An Additivity Formula......................................164
17 Tubes Formula.................................................165
17.1 The Lipschitz-Killing Curvatures.............................165
17.2 The Tubes Formula of Weyl (1939)............................168
17.2.1 The Volume of a Tube................................168
17.2.2 Intrinsic Character of the M* ..........................170
17.3 The Euler Characteristic.....................................171
17.4 Partial Continuity of the $ k..................................171
17.5 Transversal Integrals........................................172
17.6 On the Differentiability of the Immersions......................174
18 Subsets of Positive Reach .......................................177
18.1 Subsets of Positive Reach (Fédérer, 1958)......................177
18.2 The Steiner Formula........................................180
18.3 Curvature Measures ........................................182
18.4 The Euler Characteristic.....................................182
18.5 The Problem of Continuity of the í ¿..........................184
18.6 The Transversal Integrals....................................186
x Contents
Part VII The Theory of Normal Cycles
19 Invariant Forms ...............................................189
19.1 Invariant Forms onE^xE^ .................................189
19.2 Invariant Differential Forms on E^ x S^ 1 .....................190
19.3 Examples in Low Dimensions................................192
20 The Normal Cycle..............................................193
20.1 The Notion of a Normal Cycle................................193
20.1.1 Normal Cycle of a Smooth Submanifold.................194
20.1.2 Normal Cycle of a Subset of Positive Reach..............194
20.1.3 Normal Cycle of a Polyhedron.........................195
20.1.4 Normal Cycle of a Subanalytic Set......................196
20.2 Existence and Uniqueness of the Normal Cycle.................196
20.3 A Convergence Theorem....................................198
20.3.1 Boundness of the Mass of Normal Cycles................199
20.3.2 Convergence of the Normal Cycles.....................199
20.4 Approximation of Normal Cycles.............................200
21 Curvature Measures of Geometric Sets...........................205
21.1 Definition of Curvatures.....................................205
21.1.1 The Case of Smooth Submanifolds.....................206
21.1.2 The Case of Polyhedra................................208
21.2 Continuity of the Mk.......................................209
21.3 Curvature Measures of Geometric Sets.........................210
21.4 Convergence and Approximation Theorems ....................210
22 Second Fundamental Measure...................................213
22.1 A Vector-Valued Invariant Form..............................213
22.2 Second Fundamental Measure Associated to a Geometric Set......214
22.3 The Case of a Smooth Hypersurface...........................215
22.4 The Case of a Polyhedron....................................216
22.5 Convergence and Approximation .............................216
22.6 An Example of Application..................................217
Part VIII Applications to Curves and Surfaces
23 Curvature Measures in E2 ......................................221
23.1 Invariant Forms of E2 x § ...................................221
23.2 Bounded Domains in E2.....................................221
23.2.1 The Normal Cycle of a Bounded Domain................221
23.2.2 The Mass of the Normal Cycle of a Domain in E2.........223
23.3 Plane Curves..............................................224
23.3.1 The Normal Cycle of an (Embedded) Curve in E2.........224
23.3.2 The Mass of the Normal Cycle of a Curve in E2 ..........225
Contents xi
23.4 The Length of Plane Curves..................................226
23.4.1 Smooth Curves......................................226
23.4.2 Polygon Lines.......................................227
23.5 The Curvature of Plane Curves...............................227
23.5.1 Smooth Curves......................................227
23.5.2 Polygon Lines.......................................228
24 Curvature Measures in E3 ......................................231
24.1 Invariant Forms of E3 x S2...................................231
24.2 Space Curves and Polygons..................................231
24.2.1 The Normal Cycle of Space Curves.....................231
24.2.2 The Length of Space Curves...........................232
24.2.3 The Curvature of Space Curves ........................233
24.3 Surfaces and Bounded Domains in E3 .........................234
24.3.1 The Normal Cycle of a Bounded Domain................234
24.3.2 The Mass of the Normal Cycle of a Domain in E3.........235
24.3.3 The Curvature Measures of a Domain...................236
24.4 Second Fundamental Measure for Surfaces.....................238
25 Approximation of the Curvature of Curves........................241
25.1 Curves in E2...............................................241
25.2 Curves in E3...............................................242
26 Approximation of the Curvatures of Surfaces .....................249
26.1 The General Approximation Result............................249
26.2 Approximation by a Triangulation ............................250
26.2.1 A Bound on the Mass of the Normal Cycle...............250
26.2.2 Approximation of the Curvatures.......................251
26.2.3 Triangulations Closely Inscribed in a Surface.............252
27 On Restricted Delaunay Triangulations...........................253
27.1 Delaunay Triangulation .....................................253
27.1.1 Main Definitions.....................................253
27.1.2 The Empty Ball Property..............................254
27.1.3 Delaunay Triangulation Restricted to a Subset............255
27.2 Approximation Using a Delaunay Triangulation.................256
27.2.1 The Notion of s-Sample ..............................256
27.2.2 A Bound on the Hausdorff Distance.....................256
27.2.3 Convergence of the Normals...........................257
27.2.4 Convergence of Length and Area.......................258
27.2.5 Convergence of Curvatures............................258
Bibliography.......................................................261
Index.............................................................265
|
| adam_txt |
Contents
1 Introduction. 1
1.1 Two Fundamental Properties. 1
1.2 Different Possible Classifications. 2
1.3 Part I: Motivation . 3
1.4 Part II: Background - Metric and Measures. 4
1.5 Part III: Background - Polyhedra and Convex Subsets . 4
1.6 Part IV: Background - Classical Tools on Differential Geometry . 5
1.7 Part V: On Volume. 6
1.8 Part VI: The Steiner Formula. 6
1.9 Part VII: The Theory of Normal Cycles. 7
1.10 Part VIII: Applications to Curves and Surfaces. 9
Parti Motivations
2 Motivation: Curves. 13
2.1 The Length of a Curve. 13
2.1.1 The Length of a Segment and a Polygon. 13
2.1.2 The General Definition . 14
2.1.3 The Length of a C1-Curve.:. 15
2.1.4 An Obvious Convergence Result. 16
2.1.5 Warning! Negative Results. 16
2.2 The Curvature of a Curve. 17
2.2.1 The Pointwise Curvature of a Curve . 17
2.2.2 The Global (or Total) Curvature. 19
2.3 The Gauss Map of a Curve. 21
2.4 Curves in E2. 22
2.4.1 A Pointwise Convergence Result for Plane Curves . 22
2.4.2 Warning! A Negative Result on the Approximation
by Conies. 22
2.4.3 The Signed Curvature of a Smooth Plane Curve. 24
vi Contents
2.4.4 The Signed Curvature of a Plane Polygon. 26
2.4.5 Signed Curvature and Topology. 27
2.5 Conclusion. 28
3 Motivation: Surfaces. 29
3.1 The Area of a Surface. 29
3.1.1 The Area of a Piecewise Linear Surface. 29
3.1.2 The Area of a Smooth Surface. 29
3.1.3 Warning! The Lantern of Schwarz. 30
3.2 The Pointwise Gauss Curvature. 33
3.2.1 Background on the Curvatures of Surfaces. 33
3.2.2 Gauss Curvature and Geodesic Triangles . 34
3.2.3 The Angular Defect of a Vertex of a Polyhedron. 36
3.2.4 Warning! A Negative Result. 37
3.2.5 Warning! The Pointwise Gauss Curvature of a Closed
Surface. 39
3.2.6 Warning! A Negative Result Concerning
the Approximation by Quadrics. 40
3.3 The Gauss Map of a Surface. 41
3.3.1 The Gauss Map of a Smooth Surface. 41
3.3.2 The Gauss Map of a Polyhedron. 42
3.4 The Global Gauss Curvature. 43
3.5 The Volume. 44
Part II Background: Metrics and Measures
4 Distance and Projection. 47
4.1 The Distance Function. 47
4.2 The Projection Map. 49
4.3 The Reach of a Subset. 52
4.4 The Voronoi Diagrams. 55
4.5 The Medial Axis of a Subset. 55
5 Elements of Measure Theory. 57
5.1 Outer Measures and Measures. 57
5.1.1 Outer Measures. 57
5.1.2 Measures. 58
5.1.3 Outer Measures vs. Measures. 58
5.1.4 Signed Measures. 59
5.1.5 Borel Measures. 60
5.2 Measurable Functions and Their Integrals. 60
5.2.1 Measurable Functions. 60
5.2.2 Integral of Measurable Functions. 61
Contents vii
5.3 The Standard Lebesgue Measure on E^. 62
5.3.1 Lebesgue Outer Measure on R and E^. 63
5.3.2 Lebesgue Measure on R and E^. 64
5.3.3 Change of Variable. 64
5.4 Hausdorff Measures . 65
5.5 Area and Coarea Formula. 66
5.6 Radon Measures. 67
5.7 Convergence of Measures. 67
Part III Background: Polyhedra and Convex Subsets
6 Polyhedra. 71
6.1 Definitions and Properties of Polyhedra. 71
6.2 Euler Characteristic. 74
6.3 Gauss Curvature of a Polyhedron. 75
7 Convex Subsets. 77
7.1 Convex Subsets. 77
7.1.1 Definition and Basic Properties. 77
7.1.2 The Support Function . 79
7.1.3 The Volume of Convex Bodies. 80
7.2 Differential Properties of the Boundary. 81
7.3 The Volume of the Boundary of a Convex Body. 82
7.4 The Transversal Integral and the Hadwiger Theorem. 84
7.4.1 Notion of Valuation. 84
7.4.2 Transversal Integral. 85
7.4.3 The Hadwiger Theorem. 86
Part IV Background: Classical Tools in Differential Geometry
8 Differential Forms and Densities on E^. 91
8.1 Differential Forms and Their Integrals. 91
8.1.1 Differential Forms on E^ . 91
8.1.2 Integration of ^-Differential Forms on E^. 93
8.2 Densities. 94
8.2.1 Notion of Density on E^. 94
8.2.2 Integration of Densities on EN and the Associated Measure . 95
9 Measures on Manifolds. 97
9.1 Integration of Differential Forms. 97
9.2 Density and Measure on a Manifold. 98
9.3 The Fubini Theorem on a Fiber Bundle. 99
viii Contents
10 Background on Riemannian Geometry.101
10.1 Riemannian Metric and Levi-Civita Connexion.101
10.2 Properties of the Curvature Tensor.102
10.3 Connexion Forms and Curvature Forms.103
10.4 The Volume Form.103
10.5 The Gauss-Bonnet Theorem.104
10.6 Spheres and Balls.104
10.7 The Grassmann Manifolds.105
10.7.1 The Grassmann Manifold G°(N,k) .105
10.7.2 The Grassmann Manifold G(N,k).106
10.7.3 The Grassmann Manifolds AG(N,k) and AG°(N,k) .107
11 Riemannian Submanifolds.109
11.1 Some Generalities on (Smooth) Submanifolds .109
11.2 The Volume of a Submanifold.112
11.3 Hypersurfaces in E^.113
11.3.1 The Second Fundamental Form of a Hypersurface.113
11.3.2 ¿'''-Mean Curvature of a Hypersurface.114
11.4 Submanifolds in EN of Any Codimension.115
11.4.1 The Second Fundamental Form of a Submanifold.115
11.4.2 £'A-Mean Curvatures in Large Codimension.116
11.4.3 The Normal Connexion.116
11.4.4 The Gauss-Codazzi-Ricci Equations.117
11.5 The Gauss Map of a Submanifold.118
11.5.1 The Gauss Map of a Hypersurface.118
11.5.2 The Gauss Map of a Submanifold of Any Codimension----118
12 Currents.121
12.1 Basic Definitions and Properties on Currents.121
12.2 Rectifiable Currents.122
12.3 Three Theorems.124
PartV On Volume
13 Approximation of the Volume.129
13.1 The General Framework.129
13.2 A General Evaluation Theorem for the Volume.131
13.2.1 Statement of the Main Result.131
13.2.2 Proof of Theorem 38.131
13.3 An Approximation Result .133
13.4 A Convergence Theorem for the Volume.135
13.4.1 The Framework.135
13.4.2 Statement of the Theorem.137
Contents ix
14 Approximation of the Length of Curves.139
14.1 A General Approximation Result.139
14.2 An Approximation by a Polygonal Line.140
15 Approximation of the Area of Surfaces.143
15.1 A General Approximation of the Area.143
15.2 Triangulations.144
15.2.1 Geometric Invariant Associated to a Triangle.144
15.2.2 Geometric Invariant Associated to a Triangulation.145
15.3 Relative Height of a Triangulation Inscribed in a Surface .145
15.4 A Bound on the Deviation Angle .146
15.4.1 Statement of the Result and Its Consequences.146
15.4.2 Proof of Theorem 45.147
15.5 Approximation of the Area of a Smooth Surface by the Area
of a Triangulation.150
Part VI The Steiner Formula
16 The Steiner Formula for Convex Subsets.153
16.1 The Steiner Formula for Convex Bodies (1840) .153
16.2 Examples: Segments, Discs, and Balls.155
16.3 Convex Bodies in E^ Whose Boundary is a Polyhedron.158
16.4 Convex Bodies with Smooth Boundary.159
16.5 Evaluation of the Quermassintegrale by Means of Transversal
Integrals.161
16.6 Continuity of the äE *.162
16.7 An Additivity Formula.164
17 Tubes Formula.165
17.1 The Lipschitz-Killing Curvatures.165
17.2 The Tubes Formula of Weyl (1939).168
17.2.1 The Volume of a Tube.168
17.2.2 Intrinsic Character of the M* .170
17.3 The Euler Characteristic.171
17.4 Partial Continuity of the $ k.171
17.5 Transversal Integrals.172
17.6 On the Differentiability of the Immersions.174
18 Subsets of Positive Reach .177
18.1 Subsets of Positive Reach (Fédérer, 1958).177
18.2 The Steiner Formula.180
18.3 Curvature Measures .182
18.4 The Euler Characteristic.182
18.5 The Problem of Continuity of the í ¿.184
18.6 The Transversal Integrals.186
x Contents
Part VII The Theory of Normal Cycles
19 Invariant Forms .189
19.1 Invariant Forms onE^xE^ .189
19.2 Invariant Differential Forms on E^ x S^"1 .190
19.3 Examples in Low Dimensions.192
20 The Normal Cycle.193
20.1 The Notion of a Normal Cycle.193
20.1.1 Normal Cycle of a Smooth Submanifold.194
20.1.2 Normal Cycle of a Subset of Positive Reach.194
20.1.3 Normal Cycle of a Polyhedron.195
20.1.4 Normal Cycle of a Subanalytic Set.196
20.2 Existence and Uniqueness of the Normal Cycle.196
20.3 A Convergence Theorem.198
20.3.1 Boundness of the Mass of Normal Cycles.199
20.3.2 Convergence of the Normal Cycles.199
20.4 Approximation of Normal Cycles.200
21 Curvature Measures of Geometric Sets.205
21.1 Definition of Curvatures.205
21.1.1 The Case of Smooth Submanifolds.206
21.1.2 The Case of Polyhedra.208
21.2 Continuity of the Mk.209
21.3 Curvature Measures of Geometric Sets.210
21.4 Convergence and Approximation Theorems .210
22 Second Fundamental Measure.213
22.1 A Vector-Valued Invariant Form.213
22.2 Second Fundamental Measure Associated to a Geometric Set.214
22.3 The Case of a Smooth Hypersurface.215
22.4 The Case of a Polyhedron.216
22.5 Convergence and Approximation .216
22.6 An Example of Application.217
Part VIII Applications to Curves and Surfaces
23 Curvature Measures in E2 .221
23.1 Invariant Forms of E2 x §'.221
23.2 Bounded Domains in E2.221
23.2.1 The Normal Cycle of a Bounded Domain.221
23.2.2 The Mass of the Normal Cycle of a Domain in E2.223
23.3 Plane Curves.224
23.3.1 The Normal Cycle of an (Embedded) Curve in E2.224
23.3.2 The Mass of the Normal Cycle of a Curve in E2 .225
Contents xi
23.4 The Length of Plane Curves.226
23.4.1 Smooth Curves.226
23.4.2 Polygon Lines.227
23.5 The Curvature of Plane Curves.227
23.5.1 Smooth Curves.227
23.5.2 Polygon Lines.228
24 Curvature Measures in E3 .231
24.1 Invariant Forms of E3 x S2.231
24.2 Space Curves and Polygons.231
24.2.1 The Normal Cycle of Space Curves.231
24.2.2 The Length of Space Curves.232
24.2.3 The Curvature of Space Curves .233
24.3 Surfaces and Bounded Domains in E3 .234
24.3.1 The Normal Cycle of a Bounded Domain.234
24.3.2 The Mass of the Normal Cycle of a Domain in E3.235
24.3.3 The Curvature Measures of a Domain.236
24.4 Second Fundamental Measure for Surfaces.238
25 Approximation of the Curvature of Curves.241
25.1 Curves in E2.241
25.2 Curves in E3.242
26 Approximation of the Curvatures of Surfaces .249
26.1 The General Approximation Result.249
26.2 Approximation by a Triangulation .250
26.2.1 A Bound on the Mass of the Normal Cycle.250
26.2.2 Approximation of the Curvatures.251
26.2.3 Triangulations Closely Inscribed in a Surface.252
27 On Restricted Delaunay Triangulations.253
27.1 Delaunay Triangulation .253
27.1.1 Main Definitions.253
27.1.2 The Empty Ball Property.254
27.1.3 Delaunay Triangulation Restricted to a Subset.255
27.2 Approximation Using a Delaunay Triangulation.256
27.2.1 The Notion of s-Sample .256
27.2.2 A Bound on the Hausdorff Distance.256
27.2.3 Convergence of the Normals.257
27.2.4 Convergence of Length and Area.258
27.2.5 Convergence of Curvatures.258
Bibliography.261
Index.265 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Morvan, Jean-Marie |
| author_GND | (DE-588)135744229 |
| author_facet | Morvan, Jean-Marie |
| author_role | aut |
| author_sort | Morvan, Jean-Marie |
| author_variant | j m m jmm |
| building | Verbundindex |
| bvnumber | BV023425639 |
| callnumber-first | Q - Science |
| callnumber-label | QA639 |
| callnumber-raw | QA639.5 |
| callnumber-search | QA639.5 |
| callnumber-sort | QA 3639.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)166372591 (DE-599)DNB989188892 |
| dewey-full | 516.3/62 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/62 516.362 |
| dewey-search | 516.3/62 516.362 |
| dewey-sort | 3516.3 262 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02040nam a2200565 cb4500</leader><controlfield tag="001">BV023425639</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20130327 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">080731s2008 gw ad|| |||| 00||| eng d</controlfield><datafield tag="015" ind1=" " ind2=" "><subfield code="a">08,A28,1151</subfield><subfield code="2">dnb</subfield></datafield><datafield tag="016" ind1="7" ind2=" "><subfield code="a">989188892</subfield><subfield code="2">DE-101</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783540737919</subfield><subfield code="c">Pp. : EUR 85.55 (freier Pr.), sfr 139.50 (freier Pr.)</subfield><subfield code="9">978-3-540-73791-9</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="z">9780540737919</subfield><subfield code="c">Pp. : EUR 85.55 (freier Pr.), sfr 139.50 (freier Pr.)</subfield><subfield code="9">978-0-540-73791-9</subfield></datafield><datafield tag="024" ind1="3" ind2=" "><subfield code="a">9783540737919</subfield></datafield><datafield tag="028" ind1="5" ind2="2"><subfield code="a">12096879</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)166372591</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)DNB989188892</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakddb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">gw</subfield><subfield code="c">XA-DE-BE</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-824</subfield><subfield code="a">DE-703</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-188</subfield><subfield code="a">DE-83</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA639.5</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.3/62</subfield><subfield code="2">22</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.362</subfield><subfield code="2">22/ger</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 370</subfield><subfield code="0">(DE-625)143234:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">52B10</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">510</subfield><subfield code="2">sdnb</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">53C20</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">53A04</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">52C10</subfield><subfield code="2">msc</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Morvan, Jean-Marie</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)135744229</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Generalized curvatures</subfield><subfield code="c">Jean-Marie Morvan</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Berlin [u.a.]</subfield><subfield code="b">Springer</subfield><subfield code="c">2008</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XI, 266 S.</subfield><subfield code="b">Ill., graph. Darst.</subfield><subfield code="c">24 cm</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Geometry and computing</subfield><subfield code="v">2</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Literaturverz. S. 261 - 264</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Convex geometry</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Curvature</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Discrete geometry</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Krümmungsmaß</subfield><subfield code="0">(DE-588)4242871-3</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Krümmungsmaß</subfield><subfield code="0">(DE-588)4242871-3</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-3-540-73792-6</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Geometry and computing</subfield><subfield code="v">2</subfield><subfield code="w">(DE-604)BV022959018</subfield><subfield code="9">2</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016608001&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-016608001</subfield></datafield></record></collection> |
| id | DE-604.BV023425639 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:32:43Z |
| indexdate | 2024-07-09T21:18:22Z |
| institution | BVB |
| isbn | 9783540737919 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016608001 |
| oclc_num | 166372591 |
| open_access_boolean | |
| owner | DE-824 DE-703 DE-20 DE-188 DE-83 |
| owner_facet | DE-824 DE-703 DE-20 DE-188 DE-83 |
| physical | XI, 266 S. Ill., graph. Darst. 24 cm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Geometry and computing |
| series2 | Geometry and computing |
| spelling | Morvan, Jean-Marie Verfasser (DE-588)135744229 aut Generalized curvatures Jean-Marie Morvan Berlin [u.a.] Springer 2008 XI, 266 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Geometry and computing 2 Literaturverz. S. 261 - 264 Convex geometry Curvature Discrete geometry Krümmungsmaß (DE-588)4242871-3 gnd rswk-swf Krümmungsmaß (DE-588)4242871-3 s DE-604 Erscheint auch als Online-Ausgabe 978-3-540-73792-6 Geometry and computing 2 (DE-604)BV022959018 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016608001&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Morvan, Jean-Marie Generalized curvatures Geometry and computing Convex geometry Curvature Discrete geometry Krümmungsmaß (DE-588)4242871-3 gnd |
| subject_GND | (DE-588)4242871-3 |
| title | Generalized curvatures |
| title_auth | Generalized curvatures |
| title_exact_search | Generalized curvatures |
| title_exact_search_txtP | Generalized curvatures |
| title_full | Generalized curvatures Jean-Marie Morvan |
| title_fullStr | Generalized curvatures Jean-Marie Morvan |
| title_full_unstemmed | Generalized curvatures Jean-Marie Morvan |
| title_short | Generalized curvatures |
| title_sort | generalized curvatures |
| topic | Convex geometry Curvature Discrete geometry Krümmungsmaß (DE-588)4242871-3 gnd |
| topic_facet | Convex geometry Curvature Discrete geometry Krümmungsmaß |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016608001&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV022959018 |
| work_keys_str_mv | AT morvanjeanmarie generalizedcurvatures |