Citation Context

...is the intersection of the circles centered at the vertices of a triangle of the triangulation has a non-empty interior. Circle patterns on surfaces with conical singularities are considered in [25], =-=[28]-=-. These works deal with the question of existence and uniqueness of a hyperideal circle pattern with a prescribed combinatorics and intersection angles. They provide systems of linear inequalities for...

Citation Context

...e of ideally triangulated surfaces with boundary. The λ-coordinate and the coordinate introduced in this paper are quite different. Other related works are the papers of M. Schlenker [Sc], Springborn =-=[Sp]-=- and Ushijima [Us]. The relationship between the result in this paper and works of Bonahon, Penner and Schlenker is not clear to us. This deserves a further study. A fascinating question, suggested by...

Citation Context

...any works on variational principles on circle packing or circle pattern have appeared. For example, see Brägger [2], Rivin [15], Leibon [12], Chow-Luo [8], BobenkoSpringborn [5], Guo [10], Springborn =-=[16]-=-, Luo [13], Guo-Luo [11] and others.LOCAL RIGIDITY OF INVERSIVE DISTANCE CIRCLE PACKING 5 Applying a variational principle, Colin de Verdiére [7] proved Andreev-Thurston Theorem in the special case o...

Citation Context

...g with [PP93]) that discrete minimal surfaces are not only critical points for area (fixing the combinatorics), but also have other properties similar to those of smooth minimal surfaces. 184 John M. Sullivan For instance, in a conformal parameterization of a smooth minimal surface, the coordinate functions are harmonic. To interpret this for discrete surfaces, we are led to the question of when a discrete map should be considered conformal. In general this is still open. (Interesting suggestions come from the theory of circle packings, and this is an area of active research. See for instance [Ste05, BH03, Bob08, KSS06, Spr06].) However, we should certainly agree that the identity map is conformal. A polyhedral surface M comes with an embedding IdM : M → E 3 which we consider as the identity map. Indeed, we then find (following [PP93]) that M is discrete minimal if and only if IdM is discrete harmonic. Here a polyhedral map f : M → E 3 is called discrete harmonic if it is a critical point for the Dirichlet energy, written as the following sum over the triangles T of M : E(f) := ∑ T |∇fT | 2 AreaM (T ). We can view E(f) − Area f(M) as a measure of nonconformality. For the identity map, E(IdM ) = Area(M) and ∇pE(IdM ...

Citation Context

...rivative cosine law”. Luo [7] established this principle and was able to recover the energy function of Colin de Verdiére, Brägger, Rivin, Leibon. See the related work of Luo [8] [9] [10]. Springborn =-=[15]-=- used Rivin’s strategy to solve another circle pattern problem. To state Bobenko and Springborn’s theorem, let’s recall some definitions first. Suppose Σ is a closed surface with a cell decomposition....

Citation Context

...ices and edges) was introduced by Bowers and Stephenson [4] and the variational perspective was pursued by Guo in [26] (this was anticipated by Springborn’s work on volumes of hyperideal simplices in =-=[41]-=-). The multiplicative conformal structure (corresponding, in our notation, to αi = 0 for all vertices) was apparently first suggested in [40], but most of the mathematical ideas arose in work of Luo [...

Citation Context

...onal principle for Euclidean polyhedral surfaces. Since then, many other variational principles for polyhedral surfaces have been established. See for instance, [4], [2], [7], [12], [13], [14], [19], =-=[20]-=- and others. Amazingly, almost all action functionals discovered so far are related to the Schlaefli formula. The only exception is in the beautiful work of [2]. The action functional is derived from ...