Geometry & Topology

Rigidity of polyhedral surfaces, II

Ren Guo and Feng Luo

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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.

Article information

Geom. Topol., Volume 13, Number 3 (2009), 1265-1312.

Received: 5 November 2007
Accepted: 17 January 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C26: Circle packings and discrete conformal geometry 52B70: Polyhedral manifolds 58E30: Variational principles
Secondary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations 57Q15: Triangulating manifolds

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


Guo, Ren; Luo, Feng. Rigidity of polyhedral surfaces, II. Geom. Topol. 13 (2009), no. 3, 1265--1312. doi:10.2140/gt.2009.13.1265.

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