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Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics. This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field.



Discretization of Surfaces: Special Classes and Parametrizations


Surfaces from Circles

In the search for appropriate discretizations of surface theory it is crucial to preserve fundamental properties of surfaces such as their invariance with respect to transformation groups. We discuss discretizations based on Möbius-invariant building blocks such as circles and spheres. Concrete problems considered in these lectures include the Willmore energy as well as conformal and curvature-line parametrizations of surfaces. In particular we discuss geometric properties of a recently found discrete Willmore energy. The convergence to the smooth Willmore functional is shown for special refinements of triangulations originating from a curvature-line parametrization of a surface. Further we treat special classes of discrete surfaces such as isothermic, minimal, and constant mean curvature. The construction of these surfaces is based on the theory of circle patterns, in particular on their variational description.
Alexander I. Bobenko

Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples

We construct discrete solutions to a class of boundary value problems for minimal surfaces without ends, including special classes of Plateau’s problem. The boundary consists of finitely many straight line segments lying on the surface and/or planes intersecting the surface orthogonally. The discrete minimal surfaces which satisfy the given boundary conditions are built from a combinatorial parametrization, using an orthogonal circle pattern which approximates the Gauss map and a discrete duality transformation for S-isothermic surfaces.
Ulrike Bücking

Designing Cylinders with Constant Negative Curvature

We describe algorithms that can be used to interactively construct (“design”) surfaces with constant negative curvature, in particularly those that touch a plane along a closed curve and those exhibiting a cone point. Both smooth and discrete versions of the algorithms are given.
Ulrich Pinkall

On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces

It is established that there exists an intimate connection between isometric deformations of polyhedral surfaces and discrete integrable systems. In particular, Sauer’s kinematic approach is adopted to show that second-order infinitesimal isometric deformations of discrete surfaces composed of planar quadrilaterals (discrete conjugate nets) are determined by the solutions of an integrable discrete version of Bianchi’s classical equation governing finite isometric deformations of conjugate nets. Moreover, it is demonstrated that finite isometric deformations of discrete conjugate nets are completely encapsulated in the standard integrable discretization of a particular nonlinear σ-model subject to a constraint. The deformability of discrete Voss surfaces is thereby retrieved in a natural manner.
Wolfgang K. Schief, Alexander I. Bobenko, Tim Hoffmann

Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow

In this paper Bäcklund transformations for smooth and discrete Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Hashimoto or smoke-ring flow. A doubly discrete Hashimoto flow is derived and it is shown that in this case the complex curvature of the discrete curve obeys Ablovitz and Ladik’s doubly discrete NLSE. Elastic curves (curves that evolve by rigid motion under the Hashimoto flow) in the discrete and doubly discrete case are shown to be the same.
Tim Hoffmann

The Discrete Green’s Function

We first discuss discrete holomorphic functions on quad-graphs and their relation to discrete harmonic functions on planar graphs. Then, the special weights in the discrete Cauchy-Riemann (and discrete Laplace) equations are considered, coming from quasicrystalline rhombic realizations of quad-graphs. We relate these special weights to the 3D consistency (integrability) of the discrete Cauchy-Riemann equations, allowing us to extend discrete holomorphic functions to a multidimensional lattice. Discrete exponential functions are introduced and are shown to form a basis in the space of discrete holomorphic functions growing not faster than exponentially. The discrete logarithm is constructed and characterized in various ways, including an isomonodromic property. Its real part is nothing but the discrete Green’s function.
Yuri B. Suris

Curvatures of Discrete Curves and Surfaces


Curves of Finite Total Curvature

We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fáry/Milnor, Schur, Chakerian and Wienholtz.
John M. Sullivan

Convergence and Isotopy Type for Graphs of Finite Total Curvature

Generalizing Milnor’s result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result for essential subarcs of a knot.
Elizabeth Denne, John M. Sullivan

Curvatures of Smooth and Discrete Surfaces

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss-Bonnet theorem and the mean-curvature force balance equation.
John M. Sullivan

Geometric Realizations of Combinatorial Surfaces


Polyhedral Surfaces of High Genus

The construction of the combinatorial data for a surface of maximal genus with n vertices is a classical problem: The maximal genus g = ⌊1/12(n − 3)(n − 4)⌋ was achieved in the famous “Map Color Theorem” by Ringel et al. (1968). We present the nicest one of Ringel’s constructions, for the case n ≡ 7 mod 12, but also an alternative construction, essentially due to Heffter (1898), which easily and explicitly yields surfaces of genus g ∼ 1/16 n 2.
For geometric (polyhedral) surfaces in ℝ3 with n vertices the maximal genus is not known. The current record is g ∼ 1/8n log2 n, due to McMullen, Schulz & Wills (1983). We present these surfaces with a new construction: We find them in Schlegel diagrams of “neighborly cubical 4-polytopes,” as constructed by Joswig & Ziegler (2000).
Günter M. Ziegler

Necessary Conditions for Geometric Realizability of Simplicial Complexes

We associate with any simplicial complex K and any integer m a system of linear equations and inequalities. If K has a simplicial embedding in m , then the system has an integer solution. This result extends the work of Novik (2000).
Dagmar Timmreck

Enumeration and Random Realization of Triangulated Surfaces

We discuss different approaches for the enumeration of triangulated surfaces. In particular, we enumerate all triangulated surfaces with 9 and 10 vertices. We also show how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly.
Frank H. Lutz

On Heuristic Methods for Finding Realizations of Surfaces

This article discusses heuristic methods for finding realizations of oriented matroids of rank 3 and 4. These methods can be applied for the spatial embeddability problem of 2-manifolds. They have proven successful in previous realization problems in which finally only the result was published.
Jürgen Bokowski

Geometry Processing and Modeling with Discrete Differential Geometry


What Can We Measure?

In this chapter we approach the question of “ what is measurable” from an abstract point of view using ideas from geometric measure theory. As it turns out such a first-principles approach gives us quantities such as mean and Gaussian curvature integrals in the discrete setting and more generally, fully characterizes a certain class of possible measures. Consequently one can characterize all possible “ sensible” measurements in the discrete setting which may form, for example, the basis for physical simulation.
Peter Schröder

Convergence of the Cotangent Formula: An Overview

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 L 2.
Max Wardetzky

Discrete Differential Forms for Computational Modeling

This chapter introduces the background needed to develop a geometry-based, principled approach to computational modeling. We show that the use of discrete differential forms often resolves the apparent mismatch between differential and discrete modeling, for applications varying from graphics to physical simulations. Keywords. Discrete differential forms, exterior calculus, Hodge decomposition.
Mathieu Desbrun, Eva Kanso, Yiying Tong

A Discrete Model of Thin Shells

We describe a discrete model for the dynamics of thin flexible structures, such as hats, leaves, and aluminum cans, which are characterized by a curved undeformed configuration. Previously such thin-shell models required complex continuum mechanics formulations and correspondingly complex algorithms. We show that a simple shell model can be derived geometrically for triangle meshes and implemented quickly by modifying a standard cloth simulator. Our technique convincingly simulates a variety of curved objects with materials ranging from paper to metal, as we demonstrate with several examples including a comparison of a real and simulated falling hat.
Eitan Grinspun


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