ABSTRACT
The Gauss-Bonnet Theorem, which we shall soon formulate, is, as we shall shortly see, a global result. However, the route to its proof naturally begins with a local version. However, to be able to formulate it we need to introduce a number of definitions. However, in practice, be it further mathematical uses, Generalized Relativity, Computer Graphics, Sampling Theory and Imaging, one would like to produce triangulations that enjoy certain “good” geometric properties. Moreover, by an argument already repeatedly used in Calculus, if the triangulation is positively oriented, adjacent triangles will impose opposite orientations on the common edge. While this is certainly true, a stick points in two opposite directions, and we might interpret it as showing that the total curvature of a surface determines its topology. Both ways of understanding the result are, of course, correct, and various applications necessitate the use of both implications, according to the context.
