After having investigated the real conic sections and their isoptic curves in the hyperbolic plane H 2 we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane.This topic is widely investigated in the Euclidean plane E 2 (see for example [14]), but in the hyperbolic and elliptic planes there are few results (see [4], [5] and [6]). In this paper we recall the former results on isoptic curves in the hyperbolic plane geometry, and define the notion of the generalized hyperbolic angle between proper and non-proper straight lines, summarize the generalized hyperbolic conic sections classified by K. Fladt in [8] and [9] and by E. Molnár in [17]. Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections.We use for the computations the classical model which are based on the projective interpretation of the hyperbolic geometry and in this manner the isoptic curves can be visualized on the Euclidean screen of computer.

The theory of the isoptic curves is widely studied in the Euclidean plane E 2 (see [2] and [20] and the references given there). The analogous question was investigated by the authors in the hyperbolic H 2 and elliptic, but in the higher dimensional spaces there are only a few result in this topic.In [7] we gave a natural extension of the notion of the isoptic curves to the n-dimensional Euclidean space E n (n ≥ 3) which are called isoptic hypersurfaces. Now we develope an algorithm to determine the isoptic surface HP of a 3-dimensional polytop P.We will determine the isoptic surfaces for Platonic solids and for some semi-regular Archimedean polytopes and visualize them with Wolfram Mathematica.

We study the interior angle sums of translation and geodesic triangles in the universal cover of SL 2 (R) geometry. We prove that the angle sum 3 i=1 (α i ) ≥ π for translation triangles and for geodesic triangles the angle sum can be larger, equal or less than π. * Mathematics Subject Classification 2010: 52C17, 52C22, 52B15, 53A35, 51M20.

After having investigated the real conic sections and their isoptic curves in the hyperbolic plane H 2 we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane.This topic is widely investigated in the Euclidean plane E 2 (see for example [14]), but in the hyperbolic and elliptic planes there are few results (see [4], [5] and [6]). In this paper we recall the former results on isoptic curves in the hyperbolic plane geometry, and define the notion of the generalized hyperbolic angle between proper and non-proper straight lines, summarize the generalized hyperbolic conic sections classified by K. Fladt in [8] and [9] and by E. Molnár in [17]. Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections.We use for the computations the classical model which are based on the projective interpretation of the hyperbolic geometry and in this manner the isoptic curves can be visualized on the Euclidean screen of computer.

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