Approximation of conformal mappings using conformally equivalent triangular lattices

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 $\log |f^{\prime }|$ 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 $\epsilon$.

14 pages, 3 figures; v2 typos corrected, revised introduction, some proofs extended

References in corpus (2)

• Approximation of conformal mappings by circle patterns
• Discrete analytic functions on non-uniform lattices without global geometric control

Cited by in corpus (3)

• Discrete Conformal Geometry of Polyhedral Surfaces and Its Convergence
• $C^{\infty }$-convergence of conformal mappings on triangular lattices
• CMC-1 surfaces via osculating Möbius transformations between circle patterns