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

Source abstract

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