Rigidity of polyhedral surfaces, III

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This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P Bowers and K Stephenson [4] as a generalization of Andreev and Thurston's circle packing. They conjectured that inversive distance circle packings are rigid. We prove this conjecture using recent work of R Guo [9] on the variational principle associated to the inversive distance circle packing. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures that we introduced in [11], verifying a conjecture in [11]. As a consequence, we show that the discrete Laplacian operator determines a spherical polyhedral metric.

Original languageEnglish (US)
Pages (from-to)2299-2319
Number of pages21
JournalGeometry and Topology
Issue number4
StatePublished - 2011

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


  • Circle packing
  • Curvature
  • Discrete curvature
  • Polyhedral surface
  • Rigidity


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