Location of the weighted Fermat-Torricelli point on the K-plane

摘要

We obtain the equations that allow us to compute the position of the weighted Fermat-Torricelli point on the two dimensional sphere S_K~2 of constant Gaussian curvature K and on the two dimensional hyperbolic plane H_K~2 of constant Gaussian curvature K for K ＜ 0, by introducing a method of symmetrical differentiation of a length of a geodesic arc with respect to two variable lengths of geodesic arcs, respectively. The method of differentiating a length of a geodesic arc with respect to two variable lengths of geodesic arcs is a generalization of the first variation formula of the length of a geodesic arc with respect to one variable length of geodesic arc on the K -plane.