Location of the weighted Fermat–Torricelli point on the K-plane

Abstract

We prove the following fundamental property for the Fermat-Torricelli point for four non-collinear and non-coplanar points forming a tetrahedron in R 3, which states that: The three bisecting lines having as a common vertex the Fermat-Torricelli point formed by each pair of equal angles, which are seen by the opposite edges of the tetrahedron meet perpendicularly at the Fermat-Torricelli point. Furthermore, we give an alternative proof, which is different from the one obtained by Bajaj and Mehlhos for the unsolvability of the Fermat-Torricelli problem for tetrahedra in R 3 using only algebraic computations for some angles, which have as a common vertex the Fermat-Torricelli point of the tetrahedron.

Isometric embedding of a weighted Fermat-Frechet multitree for isoperimetric deformations of the boundary of a simplex to a Frechet multisimplex in the $K$-Space

“…Isometric embedding of a spanning tree up to an additive constant, [9, Proposition 6.7,pp. [49][50][51] There is a function h…”

“…. In [50] and [51], we studied the weighted Fermat-Frechet problem for N = 3 (geodesic triangles) in the two dimensional K-plane (E 2 K ) and we found the position of the weighted Fermat-Torricelli trees, which yields the weighted Fermat-Frechet (multitree) solution.…”

An analytical solution of the weighted Fermat–Torricelli problem on a unit sphere

“…Lemma 1. [10, Theorem 1], [12] If A 0 is the weighted Fermat-Torricelli point of △A 1 A 2 A 3 , then each angle α i0j is expressed as a function of w 1 , w 2 and w 3 :…”

A Sub-idle Compressor Characteristic Generation Method With Enhanced Physical Background

“…The aerodynamic for the calculation of the zero speed curve within this work (lift and drag or pressure rise and tangential force coefficients) are obtained from previous computational fluid dynamics (CFD) studies [20] that have been validated against experimental data [21] at highly negative incidence up to approximately -70 deg. The main assumption made regarding the blade loss coefficients is that at the far off-design conditions of locked rotor operation, the actual profile of the blade is of secondary importance and the main source of losses is the highly negative incidence angle.…”

“…Fiow midspan fiow fieid of an annuiar high pressure compressor (iHPC) cascade at ioci<ed rotor conditions[21] …”