Isoptic curves to parabolas 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.

Mellish theorem for generalized constant width curves

“…A construction of an isoptic introduction as a fundamental tool in our constructions (see mainly [4,6]). Moreover, some interesting facts about isoptics are contained in [8][9][10][11][12]16,18,21,22,25]. Definition 1.1.…”

On curves with circles as their isoptics

“…We would like to emphasize that all the papers in the bibliography, that is [1][2][3][4][5][6][7][8][9][10]47,[49][50][51][52][53][54][55][56], with the exception of Santaló's and Su's books, [46,48], and the paper by Cyr, [11], present a wide spectrum of results in isoptics theory and are included here for the interested reader to have a complete overview of isoptics theory.…”

Isoptic surfaces of polyhedra

“…In [8] we extended the notion of the isoptic curves to the outer (non-proper) points of the hyperbolic plane and determined the isoptic curves of the generalized conic sections.…”

“…In the hyperbolic and elliptic planes H 2 and E 2 the isoptic curves of segments and proper conic sections are determined by the authors ( [4], [5], [6]). In [8] we extended the notion of the isoptic curves to the outer (non-proper) points of the hyperbolic plane and determined the isoptic curves of the generalized conic sections.…”