We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to address the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings. 52C26, 52B10; 57M50

Abstract. We prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces. Our method is based on two new functionals-one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdière's, Brägger's and Rivin's functionals can be derived from ours.

We define a discrete Laplace-Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finiteelements Laplacian (the so called "cotan formula") except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces.Keywords Laplace operator · Delaunay triangulation · Dirichlet energy · Simplicial surfaces · Discrete differential geometry Dirichlet Energy of Piecewise Linear FunctionsLet S be a simplicial surface in 3-dimensional Euclidean space, i.e. a geometric simplicial complex in R 3 whose carrier S is a 2-dimensional submanifold, possibly with Research for this article was supported by the DFG Research Unit 565 "Polyhedral Surfaces" and the DFG Research Center MATHEON "Mathematics for key technologies" in Berlin.A.I. Bobenko ( ) · B.A. Springborn

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We define discrete constant mean curvature (cmc) surfaces in the threedimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from orthogonal ring patterns in the two-sphere and the hyperbolic plane. We present a variational principle that allows us to solve boundary value problems and to construct discrete analogues of some classical cmc surfaces. The data used for the construction is purely combinatorialthe combinatorics of the curvature line pattern. In the limit of orthogonal circle patterns we recover the theory of discrete minimal surfaces associated to Koebe polyhedra all edges of which touch a sphere. These are generalized to two-sphere Koebe nets, i.e., nets with planar quadrilateral faces and edges that alternately touch two concentric spheres.