An analytical solution of the weighted Fermat-Torricelli problem on the unit sphere

Abstract

We obtain an analytical solution for the weighted Fermat-Torricelli problem for an equilateral geodesic triangle A_1A_2A_3 which is composed by three equal geodesic arcs (sides) of length Pi/2 for given three positive unequal weights that correspond to the three vertices on a unit sphere. This analytical solution is a generalization of Cockayne's solution given in [4] for three equal weights. Furthermore, by applying the geometric plasticity principle and the spherical cosine law, we derive a necessary condition for the weighted Fermat-Torricelli point in the form of three transcedental equations with respect to the length of the geodesic arcs A_1A_1', A_2A_2'and A_3A_3'to locate the weighted Fermat-Torricelli point A_0 at the interior of a geodesic triangle A_1'A_2'A_3'on a unit sphere with sides less than Pi/2.