We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature-like quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.
"Rigidity of polyhedral surfaces, I." J. Differential Geom. 96 (2) 241 - 302, February 2014. https://doi.org/10.4310/jdg/1393424919