Tu Berlin / Schröder / Sullivan Discrete Differential Geometry

1. Auflage 2008
ISBN: 978-3-7643-8621-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 38, 341 Seiten

Reihe: Oberwolfach Seminars

This is the first book on a newly emerging field of discrete differential geometry providing an excellent way to access this exciting area. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. The carefully edited collection of essays gives a lively, multi-facetted introduction to this emerging field.

Autoren/Hrsg.

Tu Berlin, Alexander I. Bobenko

Schröder, Peter

Sullivan, John M.

Ziegler, Günter M.

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;Part I Discretization of Surfaces: Special Classes and Parametrizations;11
3.1;Surfaces from Circles;12
3.1.1;1. Why from circles?;12
3.1.2;2. Discrete Willmore energy;14
3.1.3;3. Circular nets as discrete curvature lines;26
3.1.4;4. Discrete isothermic surfaces;28
3.1.5;5. Discrete minimal surfaces and circle patterns: geometry from combinatorics;32
3.1.6;6. Discrete conformal surfaces and circle patterns;40
3.1.7;References;41
3.2;Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples;45
3.2.1;1. Introduction;45
3.2.2;2. General construction of discrete minimal surfaces;46
3.2.3;3. Construction of solutions to special boundary value problems;48
3.2.4;4. Examples;52
3.2.5;References;62
3.3;Designing Cylinders with Constant Negative Curvature;65
3.3.1;1. Smooth surfaces;65
3.3.2;2. Discrete surfaces;69
3.3.3;3. K-surfaces with a cone point;70
3.3.4;4. K-surfaces with a planar strip;72
3.3.5;5. Software;73
3.3.6;References;73
3.4;On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces;75
3.4.1;1. Introduction;75
3.4.2;2. In.nitesimal deformations of discrete surfaces;76
3.4.3;3. Finite deformations;81
3.4.4;4. In.nitesimal deformations of second order;87
3.4.5;5. Integrability of .nite deformations;93
3.4.6;References;99
3.5;Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow;102
3.5.1;1. Introduction;102
3.5.2;2. The Hashimoto .ow, the Heisenberg flow and the nonlinear Schröodinger equation;103
3.5.3;3. The Hashimoto flow, the Heisenberg flow, and the nonlinear Schrödinger equation in the discrete case;109
3.5.4;Schrödinger equation in the discrete case;109
3.5.5;4. The doubly discrete Hashimoto flow;117
3.5.6;References;121
3.6;The Discrete Green’s Function;123
3.6.1;1. Introduction: discrete harmonic and holomorphic functions;123
3.6.2;2. Rhombically embedded quad-graphs;127
3.6.3;3. 3D consistent Cauchy–Riemann equations;129
3.6.4;4. Extending discrete holomorphic functions to a multidimensional lattice;131
3.6.5;5. Discrete exponential functions;133
3.6.6;6. The discrete logarithm;135
3.6.7;7. Isomonodromic property of the discrete logarithm;137
3.6.8;8. Conclusions;138
3.6.9;References;139
4;Part II Curvatures of Discrete Curves and Surfaces;140
4.1;Curves of Finite Total Curvature;141
4.1.1;1. Length and total variation;142
4.1.2;2. Total curvature;146
4.1.3;3. First variation of length;148
4.1.4;4. Total curvature and projection;150
4.1.5;5. Schur’s comparison theorem;154
4.1.6;6. Chakerian’s packing theorem;155
4.1.7;7. Distortion;156
4.1.8;8. A projection theorem ofWienholtz;158
4.1.9;9. Curvature density;160
4.1.10;References;162
4.2;Convergence and Isotopy Type for Graphs of Finite Total Curvature;166
4.2.1;1. Introduction;166
4.2.2;2. Definitions;167
4.2.3;3. Isotopy for thick knots;168
4.2.4;4. Isotopy for graphs of finite total curvature;169
4.2.5;5. Tame and locally flat links and graphs;172
4.2.6;6. Applications to essential arcs;173
4.2.7;7. Small isotopies in a stronger sense;174
4.2.8;References;176
4.3;Curvatures of Smooth and Discrete Surfaces;178
4.3.1;1. Smooth curves, framings and integral curvature relations;178
4.3.2;2. Curvatures of smooth surfaces;180
4.3.3;3. Integral curvature relations for surfaces;181
4.3.4;4. Discrete surfaces;182
4.3.5;5. Vector bundles on polyhedral manifolds;187
4.3.6;References;188
5;Part III Geometric Realizations of Combinatorial Surfaces;192
5.1;Polyhedral Surfaces of High Genus;193
5.1.1;1. Introduction;193
5.1.2;2. Two combinatorial constructions;196
5.1.3;3. A geometric construction;204
5.1.4;References;212
5.2;Necessary Conditions for Geometric Realizability of Simplicial Complexes;216
5.2.1;1. Introduction;216
5.2.2;2. A quick walk-through;217
5.2.3;3. Obstruction theory;218
5.2.4;4. Distinguishing between simplicial maps and PL maps;222
5.2.5;5. Geometric realizability and beyond;229
5.2.6;6. Subsystems and experiments;231
5.2.7;References;233
5.3;Enumeration and Random Realization of Triangulated Surfaces;235
5.3.1;1. Introduction;235
5.3.2;2. Triangulated surfaces and their f-vectors;236
5.3.3;3. Enumeration of triangulated surfaces;238
5.3.4;4. Lexicographic enumeration;239
5.3.5;5. Random realization;244
5.3.6;References;250
5.4;On Heuristic Methods for Finding Realizations of Surfaces;254
5.4.1;1. Introduction;254
5.4.2;2. Abstract 2-manifolds that exist on the oriented matroid level;255
5.4.3;3. Stretchability of pseudoline arrangements;256
5.4.4;4. Realizability of oriented matroids in rank 4;257
5.4.5;References;258
6;Part IV Geometry Processing and Modeling with Discrete Differential Geometry;260
6.1;What Can We Measure?;261
6.1.1;1. Introduction;261
6.1.2;2. Geometric measures;262
6.1.3;3. How many points, lines, planes, . . . hit a body?;263
6.1.4;4. The intrinsic volumes and Hadwiger’s theorem;266
6.1.5;5. Steiner’s formula;267
6.1.6;6. What all this machinery tells us;269
6.1.7;References;271
6.2;Convergence of the Cotangent Formula: An Overview;272
6.2.1;1. Introduction;272
6.2.2;2. Polyhedral surfaces;273
6.2.3;3. Convergence and approximation;275
6.2.4;References;281
6.3;Discrete Differential Forms for Computational Modeling;284
6.3.1;1. Motivation;284
6.3.2;2. Relevance of forms for integration;288
6.3.3;3. Discrete differential forms;291
6.3.4;4. Operations on chains and cochains;297
6.3.5;5. Metric-dependent operators on forms;307
6.3.6;6. Interpolation of discrete forms;311
6.3.7;7. Application to Hodge decomposition;313
6.3.8;8. Other applications;315
6.3.9;9. Conclusions;318
6.3.10;References;318
6.4;A Discrete Model of Thin Shells;322
6.4.1;1. Introduction;322
6.4.2;2. Kinematics;323
6.4.3;3. Constitutive model;324
6.4.4;4. Dynamics;327
6.4.5;5. Results;328
6.4.6;6. Further reading;330
6.4.7;References;332
7;Index;335

Leseproben

Convergence of the Cotangent Formula: An Overview (p. 275-276)

Max Wardetzky

Abstract. The cotangent formula constitutes an intrinsic discretization of the Laplace- Beltrami operator on polyhedral surfaces in a finite-element sense. This note gives an overview of approximation and convergence properties of discrete Laplacians and mean curvature vectors for polyhedral surfaces located in the vicinity of a smooth surface in euclidean 3-space. In particular, we show that mean curvature vectors converge in the sense of distributions, but fail to converge in L2. Keywords. Cotangent formula, discrete Laplacian, Laplace-Beltrami operator, convergence, discrete mean curvature.

1. Introduction

There are various approaches toward a purely discrete theory of surfaces for which classical differential geometry, and in particular the notion of curvature, appears as the limit case. Examples include the theory of spaces of bounded curvature [1, 24], Lipschitz- Killing curvatures [5, 12, 13], normal cycles [6, 7, 30, 31], circle patterns and discrete conformal structures [2, 17, 26, 28], and geometric finite elements [10, 11, 15, 20, 29]. In this note we take a finite-element viewpoint, or, more precisely, a functional-analytic one, and give an overview over convergence properties of weak versions of the Laplace- Beltrami operator and the mean curvature vector for embedded polyhedral surfaces. Convergence. Consider a sequence of polyhedral surfaces fMng, embedded into euclidean 3-space, which converges (in an appropriate sense) to a smooth embedded surface M. One may ask: What are the measures and conditions such that metric and geometric objects on Mn-like intrinsic distance, area, mean curvature, Gauss curvature, geodesics and the Laplace-Beltrami operator-converge to the corresponding objects on M? To date no complete answer has been given to this question in its full generality. For example, the approach of normal cycles [6, 7], while well-suited for treating convergence of curvatures of embedded polyhedra in the sense of measures, cannot deal with convergence of elliptic operators such as the Laplacian. The finite-element approach, on the other hand, while well-suited for treating convergence of elliptic operators (cf. [10, 11]) and mean curvature vectors, has its difficulties with Gauss curvature.

Despite the differences between these approaches, there is a remarkable similarity: The famous lantern of Schwarz [27] constitutes a quite general example of what can go wrong-pointwise convergence of surfaces without convergence of their normal fields. Indeed, while one cannot expect convergence of metric and geometric properties of embedded surfaces from pointwise convergence alone, it often suffices to additionally require convergence of normals. The main technical step, to show that this is so, is the construction of a bi-Lipschitz map between a smooth surface M, embedded into euclidean 3- space, and a polyhedral surfaceMh nearby, such that the metric distortion induced by this map is bounded in terms of the Hausdorff distance between M and Mh, the deviation of normals, and the shape operator of M. (See Theorem 3.3 and compare [19] for a similar result.) This map then allows for explicit error estimates for the distortion of area and length, and-when combined with a functional-analytic viewpoint-error estimates for the Laplace-Beltrami operator and the mean curvature vector.

We treat convergence of Laplace-Beltrami operators in operator norm, and we discuss two distinct concepts of mean curvature: a functional representation (in the sense of distributions) as well as a representation as a piecewise linear function. We observe that one concept (the functional) converges whereas the other (the function) in general does not. This is in accordance with what has been observed in geometric measure theory [5, 6, 7]: for polyhedral surfaces approximating smooth surfaces, in general, one cannot expect pointwise convergence of curvatures, but only convergence in an integrated sense.