Discrete Differential Geometry (eBook)

Preface 6
Contents 9
Part I Discretization of Surfaces: Special Classes and Parametrizations 11
Surfaces from Circles 12
1. Why from circles? 12
2. Discrete Willmore energy 14
3. Circular nets as discrete curvature lines 26
4. Discrete isothermic surfaces 28
5. Discrete minimal surfaces and circle patterns: geometry from combinatorics 32
6. Discrete conformal surfaces and circle patterns 40
References 41
Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples 45
1. Introduction 45
2. General construction of discrete minimal surfaces 46
3. Construction of solutions to special boundary value problems 48
4. Examples 52
References 62
Designing Cylinders with Constant Negative Curvature 65
1. Smooth surfaces 65
2. Discrete surfaces 69
3. K-surfaces with a cone point 70
4. K-surfaces with a planar strip 72
5. Software 73
References 73
On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces 75
1. Introduction 75
2. In.nitesimal deformations of discrete surfaces 76
3. Finite deformations 81
4. In.nitesimal deformations of second order 87
5. Integrability of .nite deformations 93
References 99
Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow 102
1. Introduction 102
2. The Hashimoto .ow, the Heisenberg flow and the nonlinear Schröodinger equation 103
3. The Hashimoto flow, the Heisenberg flow, and the nonlinear Schrödinger equation in the discrete case 109
Schrödinger equation in the discrete case 109
4. The doubly discrete Hashimoto flow 117
References 121
The Discrete Green’s Function 123
1. Introduction: discrete harmonic and holomorphic functions 123
2. Rhombically embedded quad-graphs 127
3. 3D consistent Cauchy–Riemann equations 129
4. Extending discrete holomorphic functions to a multidimensional lattice 131
5. Discrete exponential functions 133
6. The discrete logarithm 135
7. Isomonodromic property of the discrete logarithm 137
8. Conclusions 138
References 139
Part II Curvatures of Discrete Curves and Surfaces 140
Curves of Finite Total Curvature 141
1. Length and total variation 142
2. Total curvature 146
3. First variation of length 148
4. Total curvature and projection 150
5. Schur’s comparison theorem 154
6. Chakerian’s packing theorem 155
7. Distortion 156
8. A projection theorem ofWienholtz 158
9. Curvature density 160
References 162
Convergence and Isotopy Type for Graphs of Finite Total Curvature 166
1. Introduction 166
2. Definitions 167
3. Isotopy for thick knots 168
4. Isotopy for graphs of finite total curvature 169
5. Tame and locally flat links and graphs 172
6. Applications to essential arcs 173
7. Small isotopies in a stronger sense 174
References 176
Curvatures of Smooth and Discrete Surfaces 178
1. Smooth curves, framings and integral curvature relations 178
2. Curvatures of smooth surfaces 180
3. Integral curvature relations for surfaces 181
4. Discrete surfaces 182
5. Vector bundles on polyhedral manifolds 187
References 188
Part III Geometric Realizations of Combinatorial Surfaces 192
Polyhedral Surfaces of High Genus 193
1. Introduction 193
2. Two combinatorial constructions 196
3. A geometric construction 204
References 212
Necessary Conditions for Geometric Realizability of Simplicial Complexes 216
1. Introduction 216
2. A quick walk-through 217
3. Obstruction theory 218
4. Distinguishing between simplicial maps and PL maps 222
5. Geometric realizability and beyond 229
6. Subsystems and experiments 231
References 233
Enumeration and Random Realization of Triangulated Surfaces 235
1. Introduction 235
2. Triangulated surfaces and their f-vectors 236
3. Enumeration of triangulated surfaces 238
4. Lexicographic enumeration 239
5. Random realization 244
References 250
On Heuristic Methods for Finding Realizations of Surfaces 254
1. Introduction 254
2. Abstract 2-manifolds that exist on the oriented matroid level 255
3. Stretchability of pseudoline arrangements 256
4. Realizability of oriented matroids in rank 4 257
References 258
Part IV Geometry Processing and Modeling with Discrete Differential Geometry 260
What Can We Measure? 261
1. Introduction 261
2. Geometric measures 262
3. How many points, lines, planes, . . . hit a body? 263
4. The intrinsic volumes and Hadwiger’s theorem 266
5. Steiner’s formula 267
6. What all this machinery tells us 269
References 271
Convergence of the Cotangent Formula: An Overview 272
1. Introduction 272
2. Polyhedral surfaces 273
3. Convergence and approximation 275
References 281
Discrete Differential Forms for Computational Modeling 284
1. Motivation 284
2. Relevance of forms for integration 288
3. Discrete differential forms 291
4. Operations on chains and cochains 297
5. Metric-dependent operators on forms 307
6. Interpolation of discrete forms 311
7. Application to Hodge decomposition 313
8. Other applications 315
9. Conclusions 318
References 318
A Discrete Model of Thin Shells 322
1. Introduction 322
2. Kinematics 323
3. Constitutive model 324
4. Dynamics 327
5. Results 328
6. Further reading 330
References 332
Index 335

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.