Analytical Solution for the Generalized Fermat-Torricelli Problem

TL;DR

This paper derives explicit analytical solutions for the generalized Fermat-Torricelli problem, including stationary points and critical values, and explores inverse problems for prescribed stationary point positions.

Contribution

It provides the first explicit formulas for stationary points in the generalized Fermat-Torricelli problem and addresses inverse problem solutions.

Findings

Explicit formulas for stationary points and critical values.

Analytical solutions for inverse problem of parameter selection.

Enhanced understanding of the geometric structure of the problem.

Abstract

We present explicit analytical solution for the problem of minimization of the function $ F(x,y)= \sum_{j=1}^3 m_j \sqrt{(x-x_j)^2+(y-y_j)^2} $, i.e. we find the coordinates of stationary point and the corresponding critical value of $ F(x,y) $ as functions of $ {m_j,x_j,y_j}_{j=1}^3 $. In addition, we also discuss inverse problem of finding such values of $ m_1,m_2,m_3 $ with the aim for the corresponding function $ F $ to posses a prescribed position of stationary point.