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Advances in Discrete Differential Geometry
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Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization
Alexander I. Bobenko; Stefan Sechelmann; Boris Springborn
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 , as hyperelliptic curves, and as 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.
Discrete Complex Analysis on Planar Quad-Graphs
Alexander I. Bobenko; Felix Günther
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.
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 by such discrete conformal maps defined on triangular lattices. In particular, let be an infinite triangulation of the plane with congruent strictly acute triangles. We scale this triangular lattice by and approximate a compact subset of the domain of with a portion of it. For small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by on the boundary. Furthermore we show that the corresponding discrete conformal (piecewise linear) maps converge to uniformly in with error of order .
Numerical Methods for the Discrete Map
Folkmar Bornemann; Alexander Its; Sheehan Olver; Georg Wechslberger
As a basic example in nonlinear theories of discrete complex analysis, we explore various numerical methods for the accurate evaluation of the discrete map 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.
A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths
Hana Kouřimská; Lara Skuppin; Boris Springborn
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 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 spacetime.
Complex Line Bundles Over Simplicial Complexes and Their Applications
Felix Knöppel; Ulrich Pinkall
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.
Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes
Wai Yeung Lam; Ulrich Pinkall
Given a triangulated region in the complex plane, a discrete vector field assigns a vector 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.
Vertex Normals and Face Curvatures of Triangle Meshes
Xiang Sun; Caigui Jiang; Johannes Wallner; Helmut Pottmann
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.
S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature
Alexander I. Bobenko; Tim Hoffmann
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.
Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization
Ulrike Bücking; Daniel Matthes
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 in and a unit normal vector field along , find an isothermic surface that contains , is normal to there, and is such that the tangent vector bisects the principal directions of curvature. First, we prove that this problem is uniquely solvable locally around each point of , provided that and 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 , 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.
On the Lagrangian Structure of Integrable Hierarchies
Yuri B. Suris; Mats Vermeeren
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.
On the Variational Interpretation of the Discrete KP Equation
Raphael Boll; Matteo Petrera; Yuri B. Suris
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 as well as on the root lattice . We prove that, on a lattice of dimension at least four, the corresponding Euler-Lagrange equations are equivalent to the dKP equation.
Six Topics on Inscribable Polytopes
Arnau Padrol; Günter M. Ziegler
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.
DGD Gallery: Storage, Sharing, and Publication of Digital Research Data
Michael Joswig; Milan Mehner; Stefan Sechelmann; Jan Techter; Alexander I. Bobenko
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.
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