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

TL;DR

This paper extends the concepts of isogonic and isodynamic points from triangles to higher-dimensional simplices, providing characterizations and a geometric description of the Fermat point calculation.

Contribution

It generalizes properties of triangle centers to simplices in higher dimensions and describes the Weiszfeld algorithm geometrically for simplices.

Findings

Characterizations of isogonic and isodynamic centers in simplices

Geometric description of the Fermat point calculation

Extension of triangle center properties to higher dimensions

Abstract

A non-equilateral triangle in a Euclidean plane has exactly two isogonic and two isodynamic points. There are a number of different but equivalent characterizations of these triangle centers. The aim of this paper is to work out characteristic properties of isogonic and isodynamic centers of simplices that can be transferred to higher dimensions. In addition, a geometric description of the Weiszfeld algorithm for calculating the Fermat point of a simplex is given.