Algorithms for the numerical determination of coordinates of analogues of Fermat-Steiner point

Abstract

There are several points arbitrarily located in space. The problem is — to determine the coordinates of a new point, the sum of all distances to all points from which is to be minimal. Initially Pierre de Fermat, Evangelista Torricelli, and Jacobi Steiner had solved this problem but only for three points on the plane and the distance function of Euclid. For more than four on the plane the exact analytical solution have not been found yet. In this paper the algorithms for numerical determination of coordinates of Fermat-Steiner point both on the plane and in three-dimensional space for arbitrary distance metric have been presented, along with the visualization of testing.