Surface Triangulation -- The Metric Approach

Abstract: We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their Haussdorff-Gromov limit. This allows us to define in a more natural way the relevant elements, constants and invariants s.a. principal directions and principal values, Gaussian and Mean curvature, etc. By a "natural way" we mean an intrinsic, discrete, metric definitions as opposed to approximating or paraphrasing the differentiable notions. In this way we hope to circumvent computational errors and, indeed, conceptual ones, that are often inherent to the classical, "numerical" approach. In this first study we consider the problem of determining the Gaussian curvature of a polyhedral surface, by using the {\em embedding curvature} in the sense of Wald (and Menger). We present two modalities of employing these definitions for the computation of Gaussian curvature.