Discrete curvature

TL;DR

This survey explores various notions of discrete curvature in polygons and polyhedral surfaces, highlighting their connections to classical theorems and convergence to smooth curvature.

Contribution

It systematically reviews discrete curvature concepts, relates them to classical theorems, and discusses convergence results bridging discrete and smooth geometries.

Findings

Discrete curvature notions mirror classical differential geometry theorems.

Discrete curvature converges to smooth curvature under certain conditions.

The survey connects discrete and smooth geometric frameworks.

Abstract

The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about the discrete curvature are stated that repeat literally classical theorems of differential and Riemannian geometry: Theorema Egregium, Gauss--Bonnet theorem, and the Chern--Gauss--Bonnet theorem among the others. Some convergence results are also mentioned: under certain assumptions the discrete curvature tends to the smooth curvature as a smooth object is approximated by polyhedral ones.