Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

Two triangle meshes are conformally equivalent if their edge lengths are related by scale factors associated to the vertices. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study the approximation of a given smooth conformal map f by such discrete conformal maps $$f^\varepsilon $$ f ε defined on triangular lattices. In particular, let T be an infinite triangulation of the plane with congruent strictly acute triangles. We scale this triangular lattice by $$\varepsilon >0$$ ε > 0 and approximate a compact subset of the domain of f with a portion of it. For $$\varepsilon $$ ε small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $$\log |f'|$$ log | f ′ | on the boundary. Furthermore we show that the corresponding discrete conformal (piecewise linear) maps $$f^\varepsilon $$ f ε converge to f uniformly in $$C^1$$ C 1 with error of order $$\varepsilon $$ ε .