$C^{\infty }$-convergence of conformal mappings on 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 $f^{\epsilon }$, defined on a part of $\epsilon T$ for $\epsilon >0$ small enough, see [U. Bücking, 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 $f$.

17 pages, 3 figures

References in corpus (1)

• Approximation of conformal mappings by circle patterns

Cited by in corpus (3)

• CMC-1 surfaces via osculating Möbius transformations between circle patterns
• A discrete version of Liouville's theorem on conformal maps
• Conformally symmetric triangular lattices and discrete $\vartheta$-conformal maps