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We introduce a novel method for the construction of discrete conformal mappings from (regions of) embedded meshes to the plane. Our approach is based on circle patterns, i.e., arrangements of circles---one for each face---with prescribed intersection angles. Given these angles the circle radii follow as the unique minimizer of a convex energy. The method has two principal advantages over earlier approaches based on discrete harmonic mappings: (1) it supports very flexible boundary conditions ranging from natural boundaries to control of the boundary shape via prescribed curvatures; (2) the solution is based on a convex energy as a function of logarithmic radius variables with simple explicit expressions for gradients and Hessians, greatly facilitating robust and efficient numerical treatment. We demonstrate the versatility and performance of our algorithm with a variety of examples.

The theory of polyhedral surfaces... In this paper, we investigate conformal discretizations of surfaces, i.e. discretizations in terms of circles and spheres, and introduce a new discrete model for minimal surfaces. See Figs. 1 and 2. In comparison with direct methods (see, in particular, [17]), leading usually to triangle meshes, the less intuitive discretizations of the present paper have essential advantages: they respect conformal properties of surfaces, possess a maximum principle, etc...

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creation, manipulation, and interpretation. Lest we get off on the wrong foot, I should caution that this is NOT twodimensional “sphere ” packing: rather than being fixed in size, our circles must adjust their radii in tightly choreographed ways if they hope to fit together in a specified pattern. In posing this as a mathematical tale, I am asking the reader for some latitude. From a tale you expect truth without all the details; you know that the storyteller will be playing with the plot and timing; you let pictures carry part of the story. We all hope for deep insights, but perhaps sometimes a simple story with a few new twists is enough—may you enjoy this tale in that spirit. Readers who wish to dig into the details can consult the “Reader’s Guide ” at the end. Once Upon a Time … From wagon wheel to mythical symbol, predating history, perfect form to ancient geometers, companion to π, the circle is perhaps the most celebrated object in mathematics.

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Abstract. Conformal geometry is in the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces- discrete surface Ricci flow, whose continuous counter part has been used in the proof of Poincaré conjecture. Continuous Ricci flow conformally deforms a Riemannian metric on a smooth surface such that the Gaussian curvature evolves like a heat diffusion process. Eventually, the Gaussian curvature becomes constant and the limiting Riemannian metric is conformal to the original one. In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. The existence and uniqueness of the solution and the convergence of the flow process are theoretically proven, and numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow are also designed. Discrete Ricci flow has broad applications in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces.

Steinitz (1906) gave a remarkably simple and explicit description of the set of all f-vectors f(P ) = (f0 , f1 , f2) of all 3-dimensional convex polytopes. His result also identifies the simple and the simplicial 3-dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2-spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects).

We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that is uniquely determined by their curvatures. 1

We characterize hyperbolic metrics on compact triangulated surfaces with boundary using a variational principle. As a consequence, a new parameterization of the Teichmüller space of compact surface with boundary is produced. In the new parameterization, the Teichmüller space becomes an open convex polytope. It is conjectured that the Weil-Petersson symplectic form can be expressed explicitly in terms of the new coordinate. 1.1. The purpose of this paper is to produce a new parameterization of the Teichmüller space of compact surface with non-empty boundary so that the lengths of the boundary components are fixed. In this new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our result can be considered as the counter-part of