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 \(\varepsilon T\), see Bücking (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 \(C^\infty \). Furthermore, we describe how the cross-ratios of the four vertices for pairs of incident triangles are related to the Schwarzian derivative of f.
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This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”.
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Bücking, U. \(C^\infty \)-Convergence of Conformal Mappings for Conformally Equivalent Triangular Lattices. Results Math 73, 84 (2018). https://doi.org/10.1007/s00025-018-0845-2
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DOI: https://doi.org/10.1007/s00025-018-0845-2