Analytical solution of the weighted Fermat-Torricelli problem for tetrahedra: The case of two pairs of equal weights

TL;DR

This paper provides an analytical solution to the weighted Fermat-Torricelli problem for tetrahedra in three-dimensional space, specifically addressing the case where two pairs of weights are equal, advancing geometric optimization methods.

Contribution

It introduces a novel analytical solution for the weighted Fermat-Torricelli problem in tetrahedra with two pairs of equal weights, filling a gap in geometric optimization literature.

Findings

Derived explicit formulas for the solution.

Validated the solution with specific geometric configurations.

Enhanced understanding of weighted geometric optimization in 3D.

Abstract

The weighted Fermat-Torricelli problem for four non-collinear and non-coplanar points in the three dimensional Euclidean Space states that: Given four non-collinear and non-coplanar points A1, A2, A3, A4 and a positive real number (weight) Bi which correspond to each point Ai, for i = 1,2,3,4, find a fifth point such that the sum of the weighted distances to these four points is minimized. We present an analytical solution for the weighted Fermat-Torricelli problem for tetrahedra in the three dimensional Euclidean Space for the case of two pairs of equal weights.