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

TL;DR

This paper derives an analytical solution for the weighted Fermat-Torricelli problem on a unit sphere with an equilateral geodesic triangle and unequal weights, extending previous solutions and providing necessary conditions for the optimal point.

Contribution

It generalizes Cockayne's solution to unequal weights and derives necessary conditions using geometric principles and spherical cosine law.

Findings

Analytical solution for weighted Fermat-Torricelli problem on a sphere.

Necessary conditions expressed as three transcendental equations.

Extension of previous solutions to unequal weights.

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.