Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in textbf{Nil} Geometry

Abstract

After having investigated the geodesic and translation triangles and their angle sums in textbf{Sol} and widetilde{{textbf{S}}{textbf{L}}_2{textbf{R}}} geometries we consider the analogous problem in textbf{Nil} space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in textbf{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 textbf{Nil} isoptic surfaces of translation-like segments and as a special case of this we examine the textbf{Nil} translation-like Thales sphere, which we call Thaloid. In our work we will use the projective model of textbf{Nil} described by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).