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

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}.