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

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 $R^{\ast }$, 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 $R^{\ast }$. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one $R^{\ast }$ is unique, in the second one three values $R^{\ast }$ there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of $R$-bodies.

24 pages, 6 figures, 3 appendices