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Geometriae Dedicata

, Volume 137, Issue 1, pp 163–197 | Cite as

Approximation of conformal mappings by circle patterns

  • Ulrike BückingEmail author
Original Paper

Abstract

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 corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in (0, π). 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 decreasing to 0 and with uniformly bounded intersection angles. The image circle pattern has the same combinatorics and intersection angles and is determined from boundary conditions (radii or angles) according to the values of g′ (|g′| or arg g′). For quasicrystallic circle patterns the convergence result is strengthened to C -convergence on compact subsets.

Keywords

Circle pattern Convergence Conformal mappings Discrete Laplacian 

Mathematics Subject Classification (2000)

30G25 30E10 52C26 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Fakultät II – Institut für MathematikTechnische Universität BerlinBerlinGermany

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