Translation-like isoptic surfaces and angle sums of translation triangles in $\NIL$ geometry

TL;DR

This paper explores the properties of translation triangles in $ ext{NIL}$ geometry, analyzing their angle sums and introducing a method to determine isoptic surfaces, including the novel concept of a Thaloid, within this Thurston geometry.

Contribution

It provides the first analysis of angle sums in $ ext{NIL}$ geometry and develops a new approach to find isoptic surfaces of translation segments in this space.

Findings

Angle sums can be greater than or equal to π in $ ext{NIL}$ geometry.

Introduced a procedure to determine equations of $ ext{NIL}$ isoptic surfaces.

Defined the $ ext{NIL}$ translation-like Thales sphere, called Thaloid.

Abstract

After having investigated the geodesic and translation triangles and their angle sums in $\SOL$ and $\SLR$ geometries we consider the analogous problem in $\NIL$ space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in $\NIL$ geometry and we provide a new approach to prove that it can be larger than or equal to $\pi$. Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of $\NIL$ isoptic surfaces of translation-like segments and as a special case of this we examine the $\NIL$ translation-like Thales sphere, which we call {\it Thaloid}. In our work we will use the projective model of $\NIL$ described by E. Moln\'ar in \cite{M97}.