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## Über dieses Buch

This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics.

The authors take a closer look at discrete models in differential

geometry and dynamical systems. Their curves are polygonal, surfaces

Nevertheless, the difference between the corresponding smooth curves,

surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture.

This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.

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## Inhaltsverzeichnis

Open Access

### Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in $$\mathbb {R}^{3}$$R3, as hyperelliptic curves, and as $$\mathbb {CP}^{1}$$CP1 modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller.

Alexander I. Bobenko, Stefan Sechelmann, Boris Springborn

Open Access

### Discrete Complex Analysis on Planar Quad-Graphs

We develop further a linear theory of discrete complex analysis on general quad-graphs, extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph leads to generalizations as well as to new proofs of previously known discrete analogs of classical theorems. New results include in particular discretizations of Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function. Another contribution is a discussion on the product of discrete holomorphic functions that is itself discrete holomorphic in a specific sense. In this paper, we focus on planar quad-graphs, but many notions and theorems can be easily adapted to discrete Riemann surfaces. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths explicit formulae for a discrete Green’s function and discrete Cauchy’s kernels are obtained. This slightly generalizes the previous results on rhombic lattices. When we further restrict to the integer lattice of a two-dimensional skew coordinate system a discrete Cauchy’s integral formulae for higher order derivatives is derived.

Alexander I. Bobenko, Felix Günther

Open Access

### Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

Two triangle meshes are conformally equivalent if their edge lengths are related by scale factors associated to the vertices. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study the approximation of a given smooth conformal map f by such discrete conformal maps $$f^\varepsilon$$fε defined on triangular lattices. In particular, let T be an infinite triangulation of the plane with congruent strictly acute triangles. We scale this triangular lattice by $$\varepsilon >0$$ε>0 and approximate a compact subset of the domain of f with a portion of it. For $$\varepsilon$$ε small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $$\log |f'|$$log|f′| on the boundary. Furthermore we show that the corresponding discrete conformal (piecewise linear) maps $$f^\varepsilon$$fε converge to f uniformly in $$C^1$$C1 with error of order $$\varepsilon$$ε.

Ulrike Bücking

Open Access

### Numerical Methods for the Discrete Map

As a basic example in nonlinear theories of discrete complex analysis, we explore various numerical methods for the accurate evaluation of the discrete map $$Z^a$$Za introduced by Agafonov and Bobenko. The methods are based either on a discrete Painlevé equation or on the Riemann–Hilbert method. In the latter case, the underlying structure of a triangular Riemann–Hilbert problem with a non-triangular solution requires special care in the numerical approach. Complexity and numerical stability are discussed, the results are illustrated by numerical examples.

Folkmar Bornemann, Alexander Its, Sheehan Olver, Georg Wechslberger

Open Access

### A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths

We provide a new proof of the elementary geometric theorem on the existence and uniqueness of cyclic polygons with prescribed side lengths. The proof is based on a variational principle involving the central angles of the polygon as variables. The uniqueness follows from the concavity of the target function. The existence proof relies on a fundamental inequality of information theory. We also provide proofs for the corresponding theorems of spherical and hyperbolic geometry (and, as a byproduct, in $$1+1$$1+1 spacetime). The spherical theorem is reduced to the Euclidean one. The proof of the hyperbolic theorem treats three cases separately: Only the case of polygons inscribed in compact circles can be reduced to the Euclidean theorem. For the other two cases, polygons inscribed in horocycles and hypercycles, we provide separate arguments. The hypercycle case also proves the theorem for “cyclic” polygons in $$1+1$$1+1 spacetime.

Hana Kouřimská, Lara Skuppin, Boris Springborn

Open Access

### Complex Line Bundles Over Simplicial Complexes and Their Applications

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of André Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise-constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension.

Felix Knöppel, Ulrich Pinkall

Open Access

### Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector $$Y_i\in \mathbb {C}$$Yi∈C to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a Möbius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gauß map and prescribed Hopf differential.

Wai Yeung Lam, Ulrich Pinkall

Open Access

### Vertex Normals and Face Curvatures of Triangle Meshes

This study contributes to the discrete differential geometry of triangle meshes, in combination with discrete line congruences associated with such meshes. In particular we discuss when a congruence defined by linear interpolation of vertex normals deserves to be called a ‘normal’ congruence. Our main results are a discussion of various definitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula.

Xiang Sun, Caigui Jiang, Johannes Wallner, Helmut Pottmann

Open Access

### S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

We introduce a novel class of s-conical nets and, in particular, study s-conical nets with constant mean curvature. Moreover we give a unified description of nets of various types: circular, conical and s-isothermic. The later turn out to be interpolating between the circular net discretization and the s-conical one.

Alexander I. Bobenko, Tim Hoffmann

Open Access

### Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization

In this article, we study an analog of the Björling problem for isothermic surfaces (that are a generalization of minimal surfaces): given a regular curve $$\gamma$$γ in $$\mathbb {R}^3$$R3 and a unit normal vector field n along $$\gamma$$γ, find an isothermic surface that contains $$\gamma$$γ, is normal to n there, and is such that the tangent vector $$\gamma '$$γ′ bisects the principal directions of curvature. First, we prove that this problem is uniquely solvable locally around each point of $$\gamma$$γ, provided that $$\gamma$$γ and n are real analytic. The main result is that the solution can be obtained by constructing a family of discrete isothermic surfaces (in the sense of Bobenko and Pinkall) from data that is read off from $$\gamma$$γ, and then passing to the limit of vanishing mesh size. The proof relies on a rephrasing of the Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its discretization which is induced from the geometry of discrete isothermic surfaces. The discrete-to-continuous limit is carried out for the Christoffel and the Darboux transformations as well.

Ulrike Bücking, Daniel Matthes

Open Access

### On the Lagrangian Structure of Integrable Hierarchies

We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice systems. We derive the multi-time Euler Lagrange equations in their full generality for hierarchies of two-dimensional systems, and construct a pluri-Lagrangian formulation of the potential Korteweg-de Vries hierarchy.

Yuri B. Suris, Mats Vermeeren

Open Access

### On the Variational Interpretation of the Discrete KP Equation

We study the variational structure of the discrete Kadomtsev-Petviashvili (dKP) equation by means of its pluri-Lagrangian formulation. We consider the dKP equation and its variational formulation on the cubic lattice $$\mathbb Z^{N}$$ as well as on the root lattice $$Q(A_{N})$$. We prove that, on a lattice of dimension at least four, the corresponding Euler-Lagrange equations are equivalent to the dKP equation.

Raphael Boll, Matteo Petrera, Yuri B. Suris

Open Access

### Six Topics on Inscribable Polytopes

We discuss six topics related to inscribable polytopes, both in dimension 3 (where the topic was started with a problem posed by Steiner in 1832) and in higher dimensions.

Open Access

### DGD Gallery: Storage, Sharing, and Publication of Digital Research Data

We describe a project, called the DGD Gallery, whose goal is to store geometric data and to make it publicly available. The DGD Gallery offers an online web service for the storage, sharing, and publication of digital research data.

Michael Joswig, Milan Mehner, Stefan Sechelmann, Jan Techter, Alexander I. Bobenko
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