Conformal mapping in linear time

TL;DR

This paper presents a method to construct a near-conformal map from any polygonal region to the unit disk efficiently, with linear time complexity relative to the number of polygon sides.

Contribution

It introduces a linear-time algorithm for constructing approximate conformal maps for polygonal regions with controllable quasiconformal distortion.

Findings

Constructs a ($1 + oldsymbol{ ext{epsilon}}$)-quasiconformal map in linear time

Provides explicit complexity bounds depending on epsilon

Applicable to any planar region bounded by a simple polygon

Abstract

Given any $\epsilon >0$ and any planar region $\Omega$ bounded by a simple n-gon $P$ we construct a ($1 + \epsilon)$-quasiconformal map between $\Omega$ and the unit disk in time $C(\epsilon)n$. One can take $ C(\epsilon) = C + C \log (1/\epsilon) \log \log (1/\epsilon)$.