Approximation of conformal mappings using conformally equivalent triangular lattices

TL;DR

This paper demonstrates that smooth conformal maps can be effectively approximated by discrete conformal maps on scaled, conformally equivalent triangle meshes, with proven convergence and error bounds.

Contribution

It introduces a method to approximate smooth conformal maps using discrete maps on scaled triangular lattices with convergence guarantees.

Findings

Discrete conformal maps converge uniformly to smooth maps as mesh size decreases.

Error in approximation is of order epsilon, the scaling factor.

Existence of conformally equivalent meshes matching boundary scale factors is proven.

Abstract

Consider discrete conformal maps defined on the basis of two conformally equivalent triangle meshes, that is edge lengths are related by scale factors associated to the vertices. Given a smooth conformal map $f$, we show that it can be approximated by such discrete conformal maps $f^\epsilon$. In particular, let $T$ be an infinite regular triangulation of the plane with congruent triangles and only acute angles (i.e.\ $<\pi/2$). We scale this tiling by $\epsilon>0$ and approximate a compact subset of the domain of $f$ with a portion of it. For $\epsilon$ small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $\log|f'|$ on the boundary. Furthermore we show that the corresponding discrete conformal maps $f^\epsilon$ converge to $f$ uniformly in $C^1$ with error of order $\epsilon$.