Circle Packings in the Unit Disc

Approximation of analytic functions with prescribed boundary conditions by circle packing maps

We use recent advances in circle packing theory to develop a constructive method for the approximation of an analytic function $F:Ω\to \bold C$ by circle packing maps providing we have only been given $Ω$, $|F'|\big|_Ω$, and the set of critical points of $F$. This extends the earlier result of Ithiel Carter and Burt Rodin [CR] for $F$ with no critical points.

Conformally symmetric circle packings. A generalization of Doyle spirals

From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are given explicitly. It is shown, that these circle packings can be viewed as discretizations of the quotient of two Airy functions. There is an interactive version of this paper that contains JAVA applets that let you experiment with the circle packings directly. It is available at ht…

Rigidity of Circle Packings with Crosscuts

Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a bounded, simply connected domain. We introduce the concept of crosscuts and investigate the rigidity of circle packings with respect to maximal crosscuts. The main result is a discrete version of an indentity theorem for analytic functions (in the spirit of Schwarz' Lemma), which has implications to uniqueness statements for discrete conformal mappings.

mated-CRT 映射的圆填充与黎曼均匀化嵌入收敛于刘维尔量子引力

The circle packing and Riemann uniformization embedding of the mated-CRT map converges to Liouville quantum gravity

本文证明 mated-CRT 映射在圆填充和黎曼均匀化嵌入下收敛于刘维尔量子引力，并推广了 He-Schramm 的收敛结果，为后续生成树加权平面映射的收敛奠定基础。

Minimal surfaces from infinitesimal deformations of circle packings

We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of Koebe type. Furthermore, every minimal surface of Koebe type can be extended naturally to a discrete minimal surface of general type. In this way, discrete minimal surfaces via Steiner's formula are unified.

圆堆积与黎曼均匀化嵌入下树加权平面地图收敛于刘维尔量子引力

The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity

本文证明树加权平面地图在圆堆积和黎曼均匀化嵌入下收敛于$\sqrt{2}$-LQG，并附带证明自然路径收敛于SLE$_8$。

随机平面图的圆打包与黎曼统一化在ergodic自相似环境中的研究

Circle packing and Riemann uniformization of random planar maps in an ergodic scale-free environment

研究证明，在满足特定条件时，随机平面图在大规模上接近其圆打包和黎曼统一化嵌入，扩展了随机游走不变性原理的应用。

Convergence of square tilings to the Riemann map

A well-known theorem of Rodin \& Sullivan, previously conjectured by Thurston, states that the circle packing of the intersection of a lattice with a simply connected planar domain $Ω$ into the unit disc $\mathbb{D}$ converges to a Riemann map from $Ω$ to $\mathbb{D}$ when the mesh size converges to 0. We prove the analogous statement when circle packings are replaced by the square tilings of Brooks et al.