An oscillatory Fermat-Torricelli tree in R^2

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.