A class of explicit solutions for the Fermat problem for tetrahedra

TL;DR

This paper introduces a new class of explicit solutions for locating the Fermat-Torricelli point in tetrahedra, extending previous theoretical solutions by identifying specific isosceles tetrahedra configurations.

Contribution

It provides a novel class of explicit solutions for the Fermat problem in tetrahedra, generalizing earlier theoretical constructions to a broader class of isosceles tetrahedra.

Findings

Identifies a class of isosceles tetrahedra with explicit Fermat point solutions.

Extends previous theoretical solutions to a broader geometric class.

Provides formulas for the Fermat point in specific tetrahedral configurations.

Abstract

We present a class of explicit solutions for the problem of minimization of the function $f(x,y,z)=\sum_{i=1}^{4}\sqrt{(x-x_{i})^2+(y-y_{i})^2+(z-z_{i})^2},$ which gives the location of the unique stationary (Fermat-Torricelli) point for four non-collinear and non-coplanar points $A_{i}=(x_{i},y_{i},z_{i}),$ determining tetrahedra, which are derived by a proper class of isosceles tetrahedra having four equal edges and two equal opposite edges. This class of explicit solutions contains Mehlhos and Glastier's explicit solutions (theoretical constructions) obtained in \cite{Mehlhos:00} and \cite{Glastier:93}, respectively.