A complete analytical treatment of the weighted Fermat–Torricelli point for a triangle

Abstract

The weighted Fermat-Torricelli problem with positive weights α, β, and γ asks for the point in the plane of a given triangle ABC that minimizes the function f (P ) = α P A + β P B + γ P C . This paper provides a complete, fully analytical, and self-contained solution to this problem. The solution starts, as is most natural, with the gradient equation ∇f = 0, and obtains all the desired results by some delicate algebraic manipulations of this equation.The Fermat-Torricelli problem asks about the point(s) P in the Euclidean plane R 2 whose distances from the vertices of a given triangle A 1 A 2 A 3 have a minimal sum. In other words, it asks about the point(s) P at which the functionattains its minimum, where X−Y stands for the Euclidean distance between X and Y . The weighted Fermat-Torricelli problem with positive weights w 1 , w 2 , and w 3 asks about the point(s) P at which the function(1) attains its minimum. When the weights w 1 , w 2 , and w 3 are equal, the problem reduces to the original problem, which will be sometimes referred to as the unweighted Fermat-Torricelli problem. Both the weighted and unweighted Fermat-Torricelli problems, together with variants and generalizations, have attracted a great amount of attention and have generated a wealth of research. Different treatments of both versions abound in the existing literature. The

The Fermat-Torricelli Problem in the Projective Plane

“…Fermat-Torricelli points have also been considered in Minkowski spaces [CG85,MSW02], Riemannian manifolds [Yan10,Afs11] and more recently even in a tropical geometry setting [LY18]. Of great practical significance, especially in operations and location research, is the generalization of Fermat-Torricelli points to so-called Steiner-Weber points, where one seeks the point that minimizes the sum of the weighted distances to n other points, e.g., see [KM97,ZZ08,HZ17].…”