On the common points of two families of $N$-spheres in the flat $N+1$ dimensional space, each of which passes through the vertexes of a given $N$-simplex

TL;DR

This paper investigates the geometric relationships between two families of N-spheres passing through vertices of given N-simplexes in Euclidean or pseudo-Euclidean space, revealing a common locus of points with specific angular properties.

Contribution

It characterizes the geometric locus of points common to pairs of N-spheres from two families, based on fixed angular relations between sphere centers and the common points.

Findings

Identifies the set of common points for pairs of N-spheres with fixed angular relations.

Derives the geometric locus of these points, including special cases when the angle is 0 or 90 degrees.

Provides a unified geometric framework for understanding sphere families passing through simplex vertices.

Abstract

Let two distinct $N$-simplexes be given in an Euclidean or pseudo-Euclidean $N+1$ dimensional space as each is defined by the coordinates of its $N+1$ vertexes. We consider the two families of $N$-spheres passing through the vertexes of the given $N$-simplexes and the set of couples of $N$-spheres (one belonging to first family and the other to the second one). The elements of this set have at least one common point; moreover, it is such that for the angle $\alpha$ between the segments connecting that point and the centers of the corresponding $N$-spheres, there holds $\cos^{2}\alpha=const$ for each of the elements of the defined set of $N$-spheres. In the present work we find the geometric place of all these common points, including the special cases when $\cos^{2}\alpha$ is equal to $0$ or $1$.