This is a slight extension of my previous note on discrete Gauss-Bonnet theorem. As mentioned in that note, this is a generalization of the well-known fact that the sum of the exterior angles of a polygon is always , which can also be regarded as a very special case of the Gauss-Bonnet theorem.

First of all we introduce a discrete version of surfaces. Let be a (-dimensional) “discrete surface” (here for examples) or a simplicial surface (with or without boundary), which for simplicity is assumed to be contained in . A discrete surface consists of finitely many “triangles” gluing along their edges. Each triangle is isometric to an ordinary planar triangle on . We call each triangle a “face”. For each face, there are exactly three edges. And for each edge, there are exactly two vertices. If two different faces intersect, we assume they either intersect at their common edge or their common vertex. Clearly the discrete notion of surfaces is very useful in computer graphics.

The main result of this post is

Theorem 1 (Discrete Gauss-Bonnet theorem)Let be the Gaussian curvature and be the geodesic curvature on a discrete surface , then

Here is the set of interior vertices and is the set of boundary vertices and is the Euler characteristic of .

We will explain the notation below the fold.

Our first goal is to generalize all the notions such as boundary, geodesic curvature (or exterior angle at an vertex of a triangle), and Gaussian curvature on a smooth surface to a discrete surface. Continue reading