Isoptic curves of conic sections in constant curvature geometries

Abstract

In this paper, we consider the isoptic curves in 2-dimensional geometries of constant curvature E2; H2; E2. The topic is widely investigated in the Euclidean plane E2, see for example [1] and [15] and the references given there. In the hyperbolic and elliptic plane (according to [18]), there are few results in this topic (see [3] and [4]). In this paper, we give a review of the known results on isoptics of Euclidean and hyperbolic curves and develop a procedure to study the isoptic curves in the hyperbolic and elliptic plane geometries and apply it to some geometric objects, e.g. proper conic sections. For the computations we use classical models based on the projective interpretation of hyperbolic and elliptic geometry and in this manner the isoptic curves can be visualized.