An oscillatory Fermat-Torricelli tree in R^2

TL;DR

This paper introduces oscillatory Fermat-Torricelli trees as a novel solution structure for the weighted Fermat-Torricelli problem in R^2, extending mechanical interpretations and providing numerical verification.

Contribution

It generalizes the mechanical solution to include oscillatory structures in the weighted Fermat-Torricelli problem for isosceles triangles with equal weights.

Findings

New oscillatory Fermat-Torricelli tree solutions identified

Mechanical model demonstrates oscillatory behavior of the solution point

Numerical example confirms the theoretical structure

Abstract

We obtain an important generalization of the mechanical solution given by S. Gueron and R. Tessler w.r. to the weighted Fermat-Torricelli problem which derives a new structure of solutions which may be called oscillatory Fermat-Torricelli trees. The weighted Fermat-Torricelli problem in R^2 states that: Given three points in R^2 and a positive real number (weight) which correspond to each point , find the point (weighted Fermat-Torricelli point) such that the sum of the weighted distances to these three points is minimized. By applying the mechanical device of Pick and Polya the oscillatory tree solution is a new solution w.r to the weighted Fermat-Torricelli problem for a given isosceles triangle with corresponding two equal weights at the vertices of the base segment. it is worth mentioning that after time t the oscillatory knot of the mechanical system passes from the weighted Fermat-Torricelli point with non zero velocity. Furthermore, we give a numerical example to verify the structure of an oscillatory Fermat-Torricelli tree for a given isosceles triangle with equal weights.