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Discrete & Computational Geometry

, Volume 61, Issue 2, pp 380–420

# On Rigidity and Convergence of Circle Patterns

Article

## Abstract

Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. We show that two locally finite circle patterns covering the unit disc are related by a hyperbolic isometry. Furthermore, we prove an analogous rigidity statement for the complex plane if all exterior intersection angles of neighboring circles are uniformly bounded away from 0. Finally, we study a sequence of two circle patterns with the same combinatorics each of which approximates a given simply connected domain. Assume that all kites are convex and all angles in the kites are uniformly bounded and the radii of one circle pattern converge to 0. Then a subsequence of the corresponding discrete conformal maps converges to a Riemann map between the given domains.

## Keywords

Circle pattern Rigidity Discrete conformal Approximation

## Mathematics Subject Classification

52C26 52C25 30E10

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## Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

## Authors and Affiliations

• Ulrike Bücking
• 1
1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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