On the Fermat-Torricelli Points of Tetrahedra and of Higher Dimensional Simplexes

Abstract

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Analytic Approach to Optimal Routing for Commercial Formation Flight

“…This intersection is the desired point P, which minimizes the sum kPAk kPBk kPCk (sufficiencies ensuring certain types of solution are explored by Shen and Tolosa [34]). An analogous result can also be observed by constructing the corresponding circumscribed circles of each of these three new equilateral triangles, creating a concurrency at the same optimal point P. Mathematical proofs for Fermat point problems of this type (both planar and spherical) are fairly abundant: For a deeper understanding of these available methods, the authors invite you to read [35][36][37][38][39].…”

“…The properties of a curved surface mean it is impossible to find a two-dimensional (2-D) Earth projection system that is isometric [46] (i.e., preserves both angles and distances). The weighted Fermat point problem has been extended to surfaces [35][36][37]39] and to even higher dimensions [38]. Most notably Zachos and Cotsiolis [35] prove Eq.…”

Isogonic and isodynamic points of a simplex in a real affine space

“…But it is not the congruence, it is the equiangularity that justifies the name isogonic point. Moreover, if is an isogonic point inside the tetrahedron Σ, then this point is the Fermat-Torricelli point of Σ, see [1]. Let us call the Fermat-Torricelli point .…”