C∞-Convergence of Conformal Mappings for Conformally Equivalent Triangular Lattices

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, see Bucking (in: Bobenko (ed) Advances in discrete differential geometry. Springer, Berlin, pp 133-149, 2016). We improve this result and show that the convergence is in fact in C-infinity. Furthermore, we describe how the cross-ratios of the four vertices for pairs of incident triangles are related to the Schwarzian derivative of f.