Isoptic surfaces of segments in $S^2\times R$ and $H^2\times R$ geometries

Abstract

In this work, we examine the isoptic surface of line segments in the $S^2\times R$ and $H^2\times R$ geometries, which are from the 8 Thurston geometries. Based on the procedure first described in [10], we are able to give the isoptic surface of any segment implicitly. We rely heavily on the calculations published in [41, 42]. 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 $S^2\times R$ and $H^2\times R$ described by E. Moln\'ar in [20].