On the configurations of four spheres supporting the vertices of a tetrahedron

TL;DR

This paper reformulates a classical geometric theorem using polynomial systems and explores configurations of four spheres supporting a tetrahedron's vertices, revealing two main classes of solutions with distinct radius properties.

Contribution

It introduces a polynomial system for tetrahedral configurations and classifies the solutions into two main categories based on sphere radii.

Findings

Triangular pyramids are divided into two classes based on sphere radius uniqueness.

In one class, the radius R* is unique; in the other, three possible values exist.

The first class includes a subclass related to R-bodies.

Abstract

A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance coordinates to the vertices of a tetrahedron $T \subset \mathbb{R}^3$ is introduced to represent the configurations of four spheres of radius $R^*$, which intersect in one point, each sphere containing three vertices of $T$ but not the fourth one. This problem is related to that of computing the largest value $R$ for which the set of vertices of $T$ is an $R$-body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius $R^*$. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one $R^*$ is unique, in the second one three values $R^*$ there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of $R$-bodies.