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

Abstract

We obtain an analytical solution of the weighted Fermat–Torricelli (wFT) problem for a specific equilateral geodesic triangle. This approach is a generalization of Cockayne’s solution given in Cockayne (Math Mag 45:216–219, 1972) for three equal weights. Furthermore, we derive a necessary condition for the wFT point in the form of three transcendental equations with respect to some specific length of geodesic arcs. This condition locates the wFT point at the interior of a geodesic triangle with side lengths less than $$\frac{\pi }{2}$$ on a unit sphere.