On the configurations of four spheres supporting the vertices of a tetrahedron - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2024

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

Résumé

A reformulation of the three circles theorem of Johnson [10] 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 tetra- hedron T ⊂ 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 [11]. 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.
Fichier principal
Vignette du fichier
paper.pdf (475.83 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04591613 , version 1 (29-05-2024)

Identifiants

  • HAL Id : hal-04591613 , version 1

Citer

Marco Longinetti, Simone Naldi. On the configurations of four spheres supporting the vertices of a tetrahedron. 2024. ⟨hal-04591613⟩
18 Consultations
15 Téléchargements

Partager

Gmail Mastodon Facebook X LinkedIn More