Isoptic surfaces of segments in $$\textbf{S}^2\!\times \!\textbf{R}$$ and $$\textbf{H}^2\!\times \!\textbf{R}$$ geometries

Abstract

In this work, we examine the isoptic surfaces of line segments in the $$\textbf{S}^2\!\times \!\textbf{R}$$ S 2 × R and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R geometries, which are from the 8 Thurston geometries. Based on the procedure first described in Csima and Szirmai (Results Math 78:194-19, 2023), we are able to give the isoptic surface of any segment implicitly. We rely heavily on the calculations published in Szirmai (Bul Acad Ştiinţe Repub Mold Mat 2:44–61, 2020; Q J Math 73:477–494, 2022). As a special case, we examine the Thales sphere in both geometries, which are called Thaloid. In our work we will use the projective model of $$\textbf{S}^2\!\times \!\textbf{R}$$ S 2 × R and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R described by Molnár (Beitr Algebra Geom 38:261–288, 1997).