An $\epsilon$-characterization of a vertex formed by two non-overlapping geodesic arcs on surfaces with constant Gaussian curvature

Abstract

We determine a positive real number (weight) which corresponds to the intersection point (vertex) of two non-overlapping geodesic arcs, which depends on the two weights which correspond to two points of these geodesicarcs, respectively, and an infinitesimal number \epsilon. As a limiting case, for \epsilon \to 0,the triad of the corresponding weights yields a degenerate weighted Fermat-Torricelli tree which coincides with these two geodesic arcs. By applying this process for a geodesic triangle on a circular cone, we derive an \epsilon characterization of conical points in R^3.