Generalized Curvatures

Front Cover
Springer Science & Business Media, May 13, 2008 - Mathematics - 266 pages
0 Reviews
The central object of this book is the measure of geometric quantities describing N a subset of the Euclidean space (E ,), endowed with its standard scalar product. Let us state precisely what we mean by a geometric quantity. Consider a subset N S of points of the N-dimensional Euclidean space E , endowed with its standard N scalar product. LetG be the group of rigid motions of E . We say that a 0 quantity Q(S) associated toS is geometric with respect toG if the corresponding 0 quantity Q[g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter ofS and the area of the convex hull ofS are quantities geometric with respect toG . But the distance from the origin O to the closest point ofS is not, 0 since it is not invariant under translations ofS. It is important to point out that the property of being geometric depends on the chosen group. For instance, ifG is the 1 N group of projective transformations of E , then the property ofS being a circle is geometric forG but not forG , while the property of being a conic or a straight 0 1 line is geometric for bothG andG . This point of view may be generalized to any 0 1 subsetS of any vector space E endowed with a groupG acting on it.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

123 Three Theorems
124
On Volume
128
Approximation of the Volume
129
132 A General Evaluation Theorem for the Volume
131
133 An Approximation Result
133
134 A Convergence Theorem for the Volume
135
1342 Statement of the Theorem
137
Approximation of the Length of Curves
139

Motivations
12
Motivation Curves
13
212 The General Definition
14
213 The Length of a CCurve
15
214 An Obvious Convergence Result
16
22 The Curvature of a Curve
17
222 The Global or Total Curvature
19
23 The Gauss Map of a Curve
21
24 Curves in E
22
243 The Signed Curvature of a Smooth Plane Curve
24
244 The Signed Curvature of a Plane Polygon
26
245 Signed Curvature and Topology
27
25 Conclusion
28
Motivation Surfaces
29
313 Warning The Lantern of Schwarz
30
32 The Pointwise Gauss Curvature
33
322 Gauss Curvature and Geodesic Triangles
34
323 The Angular Defect of a Vertex of a Polyhedron
36
324 Warning A Negative Result
37
325 Warning The Pointwise Gauss Curvature of a Closed Surface
39
326 Warning A Negative Result Concerning the Approximation by Quadrics
40
33 The Gauss Map of a Surface
41
332 The Gauss Map of a Polyhedron
42
34 The Global Gauss Curvature
43
35 The Volume
44
Background Metrics and Measures
46
Distance and Projection
47
42 The Projection Map
49
43 The Reach of a Subset
52
44 The Voronoi Diagrams
55
Elements of Measure Theory
57
512 Measures
58
514 Signed Measures
59
515 Borel Measures
60
522 Integral of Measurable Functions
61
53 The Standard Lebesgue Measure on Eᴺ
62
531 Lebesgue Outer Measure on R and Eᴺ
63
532 Lebesgue Measure on R and Eᴺ
64
54 Hausdorff Measures
65
55 Area and Coarea Formula
66
56 Radon Measures
67
Background Polyhedra and Convex Subsets
69
Polyhedra
71
62 Euler Characteristic
74
63 Gauss Curvature of a Polyhedron
75
Convex Subsets
77
712 The Support Function
79
713 The Volume of Convex Bodies
80
72 Differential Properties of the Boundary
81
73 The Volume of the Boundary of a Convex Body
82
74 The Transversal Integral and the Hadwiger Theorem
84
742 Transversal Integral
85
743 The Hadwiger Theorem
86
Background Classical Tools in Differential Geometry
90
Differential Forms and Densities on Eᴺ
91
812 Integration of NDifferential Forms on Eᴺ
93
82 Densities
94
822 Integration of Densities on Eᴺ and the Associated Measure
95
Measures on Manifolds
97
92 Density and Measure on a Manifold
98
93 The Fubini Theorem on a Fiber Bundle
99
Background on Riemannian Geometry
101
102 Properties of the Curvature Tensor
102
103 Connexion Forms and Curvature Forms
103
105 The GaussBonnet Theorem
104
107 The Grassmann Manifolds
105
1072 The Grassmann Manifold GNk
106
1073 The Grassmann Manifolds AGNk and AGᴼNk
107
Riemannian Submanifolds
109
112 The Volume of a Submanifold
112
113 Hypersurfaces in Eᴺ
113
1132 kthMean Curvature of a Hypersurface
114
114 Submanifolds in Eᴺ of Any Codimension
115
1142 kthMean Curvatures in Large Codimension
116
1144 The GaussCodazziRicci Equations
117
115 The Gauss Map of a Submanifold
118
Currents
121
122 Rectifiable Currents
122
142 An Approximation by a Polygonal Line
140
Approximation of the Area of Surfaces
143
152 Triangulations
144
1522 Geometric Invariant Associated to a Triangulation
145
154 A Bound on the Deviation Angle
146
1542 Proof of Theorem 45
147
155 Approximation of the Area of a Smooth Surface by the Area of a Triangulation
150
The Steiner Formula
151
The Steiner Formula for Convex Subsets
153
Segments Discs and Balls
155
163 Convex Bodies in Eᴺ Whose Boundary is a Polyhedron
158
164 Convex Bodies with Smooth Boundary
159
165 Evaluation of the Quermassintegrale by Means of Transversal Integrals
161
166 Continuity of the 𝜱k
162
167 An Additivity Formula
164
Tubes Formula
165
172 The Tubes Formula of Weyl 1939
168
1722 Intrinsic Character of the Mk
170
173 The Euler Characteristic
171
175 Transversal Integrals
172
176 On the Differentiability of the Immersions
174
Subsets of Positive Reach
177
182 The Steiner Formula
180
183 Curvature Measures
182
185 The Problem of Continuity of the 𝜱k
184
186 The Transversal Integrals
186
The Theory of Normal Cycles
187
Invariant Forms
189
192 Invariant Differential Forms on Eᴺ x Sᴺ
190
193 Examples in Low Dimensions
192
The Normal Cycle
193
2011 Normal Cycle of a Smooth Submanifold
194
2013 Normal Cycle of a Polyhedron
195
2014 Normal Cycle of a Subanalytic Set
196
203 A Convergence Theorem
198
2031 Boundness of the Mass of Normal Cycles
199
204 Approximation of Normal Cycles
200
Curvature Measures of Geometric Sets
205
2111 The Case of Smooth Submanifolds
206
2112 The Case of Polyhedra
208
212 Continuity of the Mk
209
213 Curvature Measures of Geometric Sets
210
Second Fundamental Measure
213
222 Second Fundamental Measure Associated to a Geometric Set
214
223 The Case of a Smooth Hypersurface
215
224 The Case of a Polyhedron
216
226 An Example of Application
217
Applications to Curves and Surfaces
220
Curvature Measures in E
221
2322 The Mass of the Normal Cycle of a Domain in E
223
233 Plane Curves
224
2332 The Mass of the Normal Cycle of a Curve in E
225
234 The Length of Plane Curves
226
2342 Polygon Lines
227
2352 Polygon Lines
228
Curvature Measures in E
231
2422 The Length of Space Curves
232
2423 The Curvature of Space Curves
233
243 Surfaces and Bounded Domains in E
234
2432 The Mass of the Normal Cycle of a Domain in E
235
2433 The Curvature Measures of a Domain
236
244 Second Fundamental Measure for Surfaces
238
Approximation of the Curvature of Curves
241
252 Curves in E
242
Approximation of the Curvatures of Surfaces
249
262 Approximation by a Triangulation
250
2622 Approximation of the Curvatures
251
2623 Triangulations Closely Inscribed in a Surface
252
On Restricted Delaunay Triangulations
253
2712 The Empty Ball Property
254
2713 Delaunay Triangulation Restricted to a Subset
255
272 Approximation Using a Delaunay Triangulation
256
2723 Convergence of the Normals
257
2724 Convergence of Length and Area
258
Bibliography
261
Index
265
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page 2 - Aegis, we feel the testing that has been done in the past, and will be done in the future...

Bibliographic information