A class of explicit solutions for the Fermat problem for tetrahedra

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.

arXiv comment14 pages, 2 figures