Rigidity of polyhedral surfaces, II

TL;DR

This paper investigates the rigidity of polyhedral surfaces through variational principles derived from cosine laws, extending previous work on rigidity and circle patterns to a more general setting.

Contribution

It introduces a unified approach to variational principles for polyhedral surfaces, including non-triangular regions, expanding the scope of prior rigidity results.

Findings

Derived new cosine laws for non-triangular regions

Established a uniform approach to variational principles

Extended rigidity results to broader classes of polyhedral surfaces

Abstract

We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a non-triangular 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 on several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko-Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.