On Geometric Property of Fermat–Torricelli Points on Sphere

Abstract

Given three points on sphere $S^2$, a point on sphere that maximizes or minimizes the sum of its Euclidean distances to the given points is called Fermat–Torricelli point. It was proved that for $A,B,C\in S^2$ and their Fermat–Torricelli point P, the distance sum $L=PA+PB+PC$ and the edges $a=BC, b=CA, c=AB$ satisfy a polynomial equation $f(L,a,b,c)=0$ of degree 12. But little is known about the geometric property of Fermat–Torricelli points, even when A, B, C are on very special positions on sphere. In this paper, we will show that for three points A, B, C on a greater circle on sphere, their Fermat–Torricelli points are either on the same greater circle or on one of four special positions (called Zeng Points) determined by A, B, C.

Supported by National Natural Science Foundation of China Grant Nos. 61772203 and 12071282.