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

Marco Longinetti, Simone Naldi

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.

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 is introduced to represent the configurations of four spheres of radius , which intersect in one point, each sphere containing three vertices of but not the fourth one. This problem is related to that of computing the largest value for which the set of vertices of is an -body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius . The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one is unique, in the second one three values there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of -bodies.
Paper Structure (23 sections, 21 theorems, 43 equations, 6 figures)

This paper contains 23 sections, 21 theorems, 43 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Paper outline.
  4. Preliminaries and Notation
  5. Basic algebraic geometry
  6. Distance geometry
  7. Double circles and spheres
  8. Plane triangles
  9. A polynomial system
  10. Description of the algebraic solutions
  11. Geometric solutions
  12. The equilateral case
  13. Tetrahedra
  14. Triangular pyramids
  15. A polynomial system
  16. ...and 8 more sections

Key Result

Theorem 1

Assume three distinct plane circles of the same radius $R$ intersect in a point $O^*$ and let $V=\{v_0,v_1,v_2\}$ be the other intersection points. Then the circle through $V$ has radius $R$ and $O^*$ is the orthocenter of $T=\text{co}(V)$.

Figures (6)

  • Figure 1: Configurations of three circles as in \ref{['johnson_theorem']} for acute-angled (left) and obtuse-angled (center) plane triangles. Four spheres satisfying property ($i$) for a simplex in $\mathbb{R}^3$ (right).
  • Figure 2: Johnson's Theorem for a plane equilateral triangle $T$.
  • Figure 3: One of the six solutions with $\rho=\frac{5}{8}$ for the regular tetrahedron.
  • Figure 4: A triangular pyramid.
  • Figure 5: Regular tetrahedron of \ref{['ex:regular']} (on the left); hemispherical pyramid of \ref{['ex:hemispherical']} (center); trirectangular pyramid of \ref{['ex:trirectangular']} (on the right).
  • ...and 1 more figures

Theorems & Definitions (34)

  • Theorem 1: Johnson johnson2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 2
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Lemma 1
  • Theorem 3
  • ...and 24 more