Interior angle sums of geodesic triangles and translation-like isoptic surfaces in Sol geometry

TL;DR

This paper investigates the interior angle sums of geodesic triangles in Sol geometry, analyzes isoptic surfaces of translation-like segments, and introduces the concept of a Sol Thaloid, expanding understanding of geometric properties in Thurston's Sol space.

Contribution

It extends the study of geodesic triangles and isoptic surfaces to Sol geometry, providing new equations and insights into their behavior in this Thurston geometry.

Findings

Interior angle sums can be greater than, less than, or equal to π.

Derived equations for Sol isoptic surfaces of translation-like segments.

Introduced and analyzed the Sol Thaloid surface.

Abstract

After having investigated the geodesic triangles and their angle sums in Nil and $Sl\times\mathbb{R}$ geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to $\pi$. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work we will use the projective model of Sol described by E. Moln\'ar in \cite{M97}.