Rigidity of Polyhedral Surfaces

TL;DR

This paper investigates the rigidity of polyhedral surfaces and their moduli space using variational principles, introducing curvature-like quantities and deriving energy functionals from the cosine law, revealing identities akin to the Bianchi identity.

Contribution

It introduces a unified variational framework for polyhedral surface rigidity, deriving new energy functionals from classical geometric laws and identities.

Findings

Curvature-like quantities determine polyhedral metrics up to isometry.

Derived energy functionals from cosine law and Legendre transformation.

Identified identities similar to the Bianchi identity in discrete geometry.

Abstract

We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law and the Lengendre transformation of them. These include energies used by Colin de Verdiere, Braegger, Rivin, Cohen-Kenyon-Propp, Leibon and Bobenko-Springborn for variational principles on triangulated surfaces. Our study is based on a set of identities satisfied by the derivative of the cosine law. These identities which exhibit similarity in all spaces of constant curvature are probably a discrete analogous of the Bianchi identity.