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

Abstract

AbstractAfter having investigated the geodesic and translation triangles and their angle sums in$$\textbf{Sol}$$Soland$$\widetilde{{\textbf{S}}{\textbf{L}}_2{\textbf{R}}}$$SL2R~geometries we consider the analogous problem in$$\textbf{Nil}$$Nilspace that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in$$\textbf{Nil}$$Nilgeometry and we provide a new approach to prove that it can be larger than or equal to$$\pi $$π. Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of$$\textbf{Nil}$$Nilisoptic surfaces of translation-like segments and as a special case of this we examine the$$\textbf{Nil}$$Niltranslation-like Thales sphere, which we callThaloid. In our work we will use the projective model of$$\textbf{Nil}$$Nildescribed by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).