Approximation of conformal mappings by circle patterns

TL;DR

This paper presents a method for approximating conformal maps using sequences of circle patterns with controlled intersection angles, demonstrating convergence properties including smooth convergence for special patterns.

Contribution

It introduces a novel approximation technique for conformal maps via circle patterns with bounded intersection angles, extending to $C^ abla$-convergence for quasicrystallic patterns.

Findings

Convergence of circle pattern approximations to conformal maps.

Extension to $C^ abla$-convergence for quasicrystallic patterns.

Method applicable to boundary value problems in conformal mapping.

Abstract

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in $(0,\pi)$. Two sequences of circle patterns are employed to approximate a given conformal map $g$ and its first derivative. For the domain of $g$ we use embedded circle patterns where all circles have the same radius decreasing to 0 and which have uniformly bounded intersection angles. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of $g'$ ($|g'|$ or $\arg g'$). For quasicrystallic circle patterns the convergence result is strengthened to $C^\infty$-convergence on compact subsets.