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Bobenko - Advances in Discrete Differential Geometry

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Bobenko Advances in Discrete Differential Geometry
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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 -- Discrete complex analysis on planar quad-graphs: Alexander I. Bobenko and Felix Gnther -- Approximation of conformal mappings using conformally equivalent triangular lattices: Ulrike Bcking -- Numerical Methods for the Discrete Map Za: Folkmar Bornemann, Alexander Its, Sheehan Olver, and Georg Wechslberger -- A variational principle for cyclic polygons with prescribed edge lengths: Hana Kourimsk, Lara Skuppin, Boris Springborn -- Complex Line Bundles over Simplicial Complexes and their Applications: Felix Knppel and Ulrich Pinkall -- Holomorphic vector fields and quadratic differentials on planar triangular meshes: Wai Yeung Lam, Ulrich Pinkall -- Vertex normals and face curvatures of triangle meshes: Xiang Sun, Caigui Jiang, Johannes Wallner, and Helmut Pottmann -- S-conical cmc surfaces. Towards a unified theory of discrete surfaces with constant mean curvature: Alexander I. Bobenko and Tim Hoffmann -- Constructing solutions to the Bjrling problem for isothermic surfaces by structure preserving discretization: Ulrike Bcking and Daniel Matthes -- On the Lagrangian Structure of Integrable Hierarchies: Yuri B. Suris, Mats Vermeeren -- On the variational interpretation of the discrete KP equation: Raphael Boll, Matteo Petrera, and Yuri B. Suris -- Six topics on inscribable polytopes: Arnau Padrol and Gnter M. Ziegler -- DGD Gallery: Storage, sharing, and publication of digital research data: Michael Joswig, Milan Mehner, Stefan Sechelmann, Jan Techter, and Alexander I. Bobenko.;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 are made from triangles and quadrilaterals, and time is discrete. 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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The Author(s) 2016
Alexander I. Bobenko (ed.) Advances in Discrete Differential Geometry 10.1007/978-3-662-50447-5_1
Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization
Alexander I. Bobenko 1
(1)
Inst. fr Mathematik, Technische Universitt Berlin, Strae des 17. Juni 136, 10623 Berlin, Germany
Alexander I. Bobenko
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Stefan Sechelmann
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Boris Springborn (Corresponding author)
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Abstract
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 Picture 1 , as hyperelliptic curves, and as Picture 2 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.
Introduction
Not one, but several sensible definitions of discrete holomorphic functions and discrete conformal maps are known today. The oldest approach, which goes back to the early finite element literature, is to discretize the CauchyRiemann equations [].
Fig 1 Uniformization of compact Riemann surfaces The uniformization of - photo 3
Fig. 1
Uniformization of compact Riemann surfaces. The uniformization of spheres is treated in Sect. is concerned with surfaces of higher genus
The history of nonlinear theories of discrete conformal maps goes back to Thurston, who introduced patterns of circles as elementary geometric way to visualize hyperbolic polyhedra [].
A related but different nonlinear theory of discrete conformal maps is based on a straightforward definition of discrete conformal equivalence for triangulated surfaces: Two triangulations are discretely conformally equivalent if the edge lengths are related by scale factors assigned to the vertices. This also leads to a surprisingly rich theory [).
We extend the notion of discrete conformal equivalence from triangulated surfaces to polyhedral surfaces with faces that are inscribed in circles. The basic definitions and their immediate consequences are discussed in Sect..
In Sect. ).
Section is concerned with the special case of quadrilateral meshes. We discuss the emergence of orthogonal circle patterns, a peculiar necessary condition for the existence of solutions for boundary angle problems, and we extend the method of constructing discrete Riemann maps from triangulations to quadrangulations.
In Sect. , we briefly discuss discrete conformal maps from multiply connected domains to circle domains, and special cases in which we can map to slit domains.
Section deals with conformal mappings onto the sphere. We generalize the method for triangulations to quadrangulations, and we explain how the spherical version of the variational principle can in some cases be used for numerical calculations although the corresponding functional is not convex.
Section is concerned with the uniformization of tori, i.e., the representation of Riemann surfaces as a quotient space of the complex plane modulo a period lattice. We consider Riemann surfaces represented as immersed surfaces in Picture 4 , and as elliptic curves. We conduct numerical experiments to test the conjectured convergence of discrete conformal maps. We consider the difference between the true modulus of an elliptic curve (which can be calculated using hypergeometric functions) and the modulus determined by discrete uniformization, and we estimate the asymptotic dependence of this error on the number of vertices.
In Sect. , we consider the Fuchsian uniformization of Riemann surfaces represented in different forms. We consider immersed surfaces in Picture 5 (and Picture 6 ), hyperelliptic curves, and Riemann surfaces represented as a quotient of Picture 7 modulo a classical Schottky group. That is, we convert from Schottky uniformization to Fuchsian uniformization. The section ends with two extended examples demonstrating, among other things, a remarkable geometric characterization of hyperelliptic surfaces due to Schmutz Schaller.
Discrete Conformal Equivalence of Cyclic Polyhedral Surfaces
2.1 Cyclic Polyhedral Surfaces
A euclidean polyhedral surface is a surface obtained from gluing euclidean polygons along their edges. (A surface is a connected two-dimensional manifold, possibly with boundary.) In other words, a euclidean polyhedral surface is a surface equipped with, first, an intrinsic metric that is flat except at isolated points where it has cone-like singularities, and, second, the structure of a CW complex with geodesic edges. The set of vertices contains all cone-like singularities. If the surface has a boundary, the boundary is polygonal and the set of vertices contains all corners of the boundary.
Hyperbolic polyhedral surfaces and spherical polyhedral surfaces are defined analogously. They are glued from polygons in the hyperbolic and elliptic planes, respectively. Their metric is locally hyperbolic or spherical, except at cone-like singularities.
We will only be concerned with polyhedral surfaces whose faces are all cyclic, i.e., inscribed in circles. We call them cyclic polyhedral surfaces . More precisely, we require the polygons to be cyclic before they are glued together. It is not required that the circumcircles persist after gluing; they may be disturbed by cone-like singularities. A polygon in the hyperbolic plane is considered cyclic if it is inscribed in a curve of constant curvature. This may be a circle (the locus of points at constant distance from its center), a horocycle, or a curve at constant distance from a geodesic.
A triangulated surface , or triangulation for short, is a polyhedral surface all of whose faces are triangles. All triangulations are cyclic.
2.2 Notation
We will denote the sets of vertices, edges, and faces of a CW complex Picture 8 by Picture 9 , Picture 10
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