On face angles of tetrahedra with a given base

TL;DR

This paper investigates the set of face angles in tetrahedra with a fixed base, characterizing the possible cosine values of angles at the apex and describing the boundary of this set.

Contribution

It provides a detailed analysis of the geometric constraints on face angles in tetrahedra with a given base, determining the closure and boundary of the angle set.

Findings

Characterization of the closure of the set of face angles.

Description of the boundary of the set in the cosine space.

Conditions under which certain face angles are attainable.

Abstract

Let us consider the set $\Omega (\triangle ABC)$ of all tetrahedra $ABCD$ with a given non-degenerate base $ABC$ in $\mathbb{E}^3$ and $D$ lying outside the plane $ABC$. Let us denote by $\Sigma(\triangle ABC)$ the set $\left\{\Bigl(\cos \overline{\alpha},\cos \overline{\beta},\cos \overline{\gamma} \Bigr)\in \mathbb{R}^3\,|\, ABCD \in \Omega (\triangle ABC)\right\}$, where $\overline{\alpha}=\angle BDC$, $\overline{\beta}=\angle ADC$, and $\overline{\gamma}=\angle ADB$. The paper is devoted to the problem of determining of the closure of $\Sigma(\triangle ABC)$ in $\mathbb{R}^3$ and its boundary.