Rigidity of polyhedral surfaces, II

Guo, Ren¹ ; Luo, Feng²

¹ School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
² The Center of Mathematical Science, Zhejiang University, Hangzhou, Zhejiang 310027, China, and, Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

Geometry & topology, Tome 13 (2009) no. 3, pp. 1265-1312

Résumé

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.

A correction was submitted on 04 Dec 2024 and posted online on 25 Aug 2025.

Détail

DOI : 10.2140/gt.2009.13.1265

Keywords: derivative cosine law, energy function, variational principle, edge invariant, circle packing metric, circle pattern metric, polyhedral surface, rigidity, metric, curvature

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