# $C^\infty$-convergence of conformal mappings on triangular lattices

## TL;DR

This paper proves that discrete conformal maps on triangular lattices converge smoothly to smooth conformal maps, and relates cross-ratios of lattice triangles to the Schwarzian derivative of the conformal map.

## Contribution

It improves existing convergence results by establishing $C^
abla$-convergence of discrete conformal maps to smooth conformal maps on triangular lattices.

## Key findings

- Discrete conformal maps converge in $C^
abla$ to smooth conformal maps.
- Cross-ratios relate to the Schwarzian derivative of the conformal map.
- Enhanced approximation of conformal maps using triangular lattices.

## Abstract

Two triangle meshes are conformally equivalent if for any pair of incident triangles the absolute values of the corresponding cross-ratios of the four vertices agree. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study discrete conformal maps which are defined on parts of a triangular lattice $T$ with strictly acute angles. That is, $T$ is an infinite triangulation of the plane with congruent strictly acute triangles. A smooth conformal map $f$ can be approximated on a compact subset by such discrete conformal maps $f^\varepsilon$, defined on a part of $\varepsilon T$ for $\varepsilon>0$ small enough, see [U. B\"ucking, Approximation of conformal mappings using conformally equivalent triangular lattices, in "Advances in Discrete Differential Geometry" (A.I. Bobenko ed.), Springer (2016), 133--149]. We improve this result and show that the convergence is in fact in $C^\infty$. Furthermore, we describe how the cross-ratios of the four vertices for pairs of incident triangles are related to the Schwarzian derivative of $f$.