Approximation of conformal mappings by circle patterns

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G there corresponds a circle. If two vertices are connected by an edge in G then the corresponding circles intersect with an intersection angle in (0, π) and these intersection points can be associated to the dual graph G. Two sequences of circle patterns are employed to approximate a given conformal map g and its first derivative. For the domain of g we use embedded circle patterns where all circles have the same radius εn > 0 for a sequence εn → 0 and where the intersection angles are uniformly bounded. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of g (|g| or arg g). The error is of order 1/ √ − log εn. For quasicrystallic circle patterns the convergence result is strengthened to C-convergence on compact subsets and an error of order εn.