Curvature measures for discrete surfaces

The curvatures of a smooth curve or surface are local measures of its shape. Here we consider analogous measures for discrete curves and surfaces, meaning polygonal curves and triangulated polyhedral surfaces. We find that the most useful analogs are those which preserve integral relations for curvature, like the Gaus-Bonnet theorem. For simplicity, we usually restrict our attention to curves and surfaces in euclidean space R3, although many of the results would easily generalize to other ambient manifolds of arbitrary dimension.