Discrete & Computational Geometry

, Volume 57, Issue 2, pp 431–469 | Cite as

Discrete Uniformization of Polyhedral Surfaces with Non-positive Curvature and Branched Covers over the Sphere via Hyper-ideal Circle Patterns

  • Alexander I. Bobenko
  • Nikolay Dimitrov
  • Stefan Sechelmann


With the help of hyper-ideal circle pattern theory, we develop a discrete version of the classical uniformization theorems for closed polyhedral surfaces with non-positive curvature and for surfaces represented as finite branched covers over the Riemann sphere. We show that in these cases discrete uniformization via hyper-ideal circle patterns always exists and is unique. We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.


Hyper-ideal circle pattern Discrete uniformization Discrete conformal map Branched cover Polyhedral surface Variational principle 

Mathematics Subject Classification

52C26 57M50 57M12 52B10 



Research supported by Deutsche Forschungsgemeinschaft in the frame of Sonderforschungsbereich Transregio 109 “Discretization in Geometry and Dynamics”.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Alexander I. Bobenko
    • 1
  • Nikolay Dimitrov
    • 1
  • Stefan Sechelmann
    • 1
  1. 1.Institut für Mathematik MA 8-3Technische Universität BerlinBerlinGermany

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