Curvature on Discrete Surfaces

Abstract

We explore the notions of Gaussian curvature and mean curvature on discrete surfaces. In the introduction, we define discrete surfaces rigorously and introduce relevant notions of curvature on smooth surfaces. Then, in section 2, we use the local version of the smooth Gauss-Bonnet theorem to define Gaussian curvature on discrete surfaces. Using this discrete Gaussian curvature, we are able to find a discrete analogy for the global Gauss-Bonnet theorem. Next, in section 3, we use a discrete analogy of the first variation of area to define mean curvature on discrete surfaces. We are then able to define a discrete minimal surface as one that is critical for area amongst continuous piecewise linear variations of discrete surfaces which preserve the boundary conditions. Finally, we consider an explicit example of a discrete catenoid which is a discrete minimal surface and which converges to a smooth catenoid in increasing the number of vertices in the discrete surface.