Rigidity of polyhedral surfaces, II
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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.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/47bad49c6ca6e4c807333c793cf6d9d1aff931c6
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Local rigidity of inversive distance circle packing (2009, background): “In fat, that M1 is positive definite was proved in the paper Guo-Luo [ 12 ] (Lemma 4.4), by showing that its leading principal 1 × 1 and 2 × 2 minor are positive.”
- A discrete uniformization theorem for polyhedral surfaces II (2014, background): “See also [2], [10] and [9] for a discussion of Delaunay triangulations of decorated metrics.”