Isoptic Curves of Generalized Conic Sections in the Hyperbolic Plane

Abstract

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.

To a question on the existence of regular mosaics on a hyperbolic plane of positive curvature

“…Let us address, for example, Definition 2.1 of generalized angles on the hyperbolic plane from paper [13]. To reduce reasonings, we shall consider only objects of the plane O H .…”

“…A hyperbolic angle and a hyperbolic pseudoangle are topologically distinct. In article [13] these objects are not distinguishable and geometrically are not defined. It is worth noticing that the phrase "their angle is the length" from Definition 2.1 is incorrect because an angle is a geometrical figure, and the length is a number.…”

“…4, b). These topologically different objects are not distinguishable and geometrically are not defined in paper [13] too. This means that at the using of Definition 2.1 the questions about polygons angles in the plane O H will remain unresolved.…”

“…This means that at the using of Definition 2.1 the questions about polygons angles in the plane O H will remain unresolved. Formally Formula (6) from [13] yields the correct result only for the hyperbolic cosine of the measures of hyperbolic or elliptic angles and hyperbolic or elliptic pseudoangles. But Formula (6) does not give the actual values of the measures of these objects.…”

“…Moreover, Remark 2.2 conclusively confuses the reader. The hyperbolic measures of adjacent or, in terms of paper [13], complementary angles except quasiangles are differ in their nature. The measure of a hyperbolic or elliptic angle is real, the measure of its adjacent pseudoangle is complex (see Table 1).…”