Rigidity of polyhedral surfaces, II

Ren Guo, Feng Luo

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


We study the rigidity of polyhedral surfaces using variational principles. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.

Original languageEnglish (US)
Pages (from-to)1265-1312
Number of pages48
JournalGeometry and Topology
Issue number3
StatePublished - 2009

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


  • Circle packing metric
  • Circle pattern metric
  • Curvature
  • Derivative cosine law
  • Edge invariant
  • Energy function
  • Metric
  • Polyhedral surface
  • Rigidity
  • Variational principle


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