Bisector surfaces and circumscribed spheres of tetrahedra derived by translation curves in $\SOL$ geometry

TL;DR

This paper explores bisector surfaces and circumscribed spheres of tetrahedra in the $ ext{SOL}$ geometry, providing formulas and methods for their determination using the projective model, and analyzing properties of translation triangles.

Contribution

It introduces explicit equations for bisector surfaces and a method to find circumscribed spheres of tetrahedra in $ ext{SOL}$ geometry, expanding understanding of geometric structures in this space.

Findings

Bisector surfaces of two points in $ ext{SOL}$ are explicitly characterized.

The isosceles property in translation triangles does not imply equal angles.

A method for determining circumscribed translation spheres of tetrahedra is developed.

Abstract

In the present paper we study the $\SOL$ geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. Moreover, we develop a method to determine the centre and the radius of the circumscribed translation sphere of a given {\it translation tetrahedron}. In our work we will use for computations and visualizations the projective model of $\SOL$ described by E. Moln\'ar in \cite{M97}.