R-hulloid of the vertices of a tetrahedron

TL;DR

This paper studies the geometric properties of the R-hulloid of a tetrahedron's vertices in 3D space, exploring conditions for a special collapsing point and generalizing planar results to three dimensions.

Contribution

It introduces the concept of R-hulloid for tetrahedron vertices, analyzes the existence of a collapsing point, and extends planar geometric results to three dimensions.

Findings

Existence of a critical radius R* where subsets collapse into a point

Characterization of the range of ρ for V to be a ρ-body

Generalization of planar three circles theorem to 3D tetrahedral case

Abstract

The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, of the set of vertices $V$ of a tetrahedron $T$ is the minimal closed set containing $V$ such that its complement is the union of open balls of radius $R$. When $R$ is greater than the circumradius of $T$, the boundary of the $R$-hulloid consists of $V$ and possibly of four spherical subsets of well defined spheres of radius $R$ through the vertices of $T$. The existence of a value $R^*$ such that these subsets collapse into a point $O^*$, in the interior of $T$, is investigated; in such a case $O^*$ belongs to four spheres of radius $R^*$, each one through three vertices of $T$ and not containing the fourth one. As a consequence, the range of $\rho$ such that $V$ is a $\rho$-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem.