Isoptic Curves of Generalized Conic Sections in the Hyperbolic Plane

Abstract

We recall the notion of generalized hyperbolic angle between proper and improper straight lines, which is only available in Hungarian and Esperanto. Then we summarize the generalized hyperbolic conic sections. After investigation of the real conic sections and their isoptic curves in the hyperbolic plane $\mathbf{H}^2,$ we consider the problem of isoptic curves of generalized conic sections in the extended hyperbolic plane. This problem is widely investigated in the Euclidean plane $\mathbf{E}^2$ but, in the hyperbolic and elliptic planes, there are few results. Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections. For our computations, we use the classical models based on the projective interpretation of hyperbolic geometry. In this way, the isoptic curves can be visualized in the Euclidean screen of a computer.