Discrete & Computational Geometry

, Volume 61, Issue 2, pp 380–420

# On Rigidity and Convergence of Circle Patterns

• Ulrike Bücking
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

## Notes

### Acknowledgements

The author is grateful to Boris Springborn for useful discussions and advice. The author would also like to thank the anonymous referees for their valuable comments which helped to improve the manuscript.

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