On Rigidity and Convergence of Circle Patterns View Full Text


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Article Info

DATE

2019-03

AUTHORS

Ulrike Bücking

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. More... »

PAGES

1-41

References to SciGraph publications

  • 1995-09. Hyperbolic and parabolic packings in DISCRETE & COMPUTATIONAL GEOMETRY
  • 1996-06. On the convergence of circle packings to the Riemann map in INVENTIONES MATHEMATICAE
  • 1992-12. How to cage an egg in INVENTIONES MATHEMATICAE
  • 2008-12. Approximation of conformal mappings by circle patterns in GEOMETRIAE DEDICATA
  • 1993-02. Square tilings with prescribed combinatorics in ISRAEL JOURNAL OF MATHEMATICS
  • 1973. Quasiconformal Mappings in the Plane in NONE
  • 1994-09. The combinatorial Riemann mapping theorem in ACTA MATHEMATICA
  • 1998-09. TheC∞-convergence of hexagonal disk packings to the Riemann map in ACTA MATHEMATICA
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00454-018-0022-0

    DOI

    http://dx.doi.org/10.1007/s00454-018-0022-0

    DIMENSIONS

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