Hyperbolic Raisa Orbits of the Second Order in an Extended Hyperbolic Plane

Yıl 2021, Cilt: 14 Sayı: 1, 194 - 206, 15.04.2021

Lyudmila N. Romakina

https://doi.org/10.36890/iejg.904467

Öz

In this paper, we study conics, which are invariant under the hyperbolic inversion with respect to the absolute of an extended hyperbolic plane $H^2$ of curvature radius $\rho$, $\rho \in \mathbb R_+$. They are called the hyperbolic Raisa Orbits of the second order. We prove that each hyperbolic Raisa Orbits of the second order in $H^2$ belongs to one of four conics types of this plane. These types are as follows: the bihyperbolas of one sheet; the hyperbolas; the hyperbolic parabolas of one sheet and two branches; the elliptic cycles of radius $\pi \rho / 4$. The family of all hyperbolic Raisa Orbits from the family of all bihyperbolas of one sheet (or all hyperbolas) defined exactly up to motions, is one-parametric. The family of all hyperbolic Raisa Orbits from the family of all hyperbolic parabolas of one sheet and two branches (or all elliptic cycles) contains a unique conic defined exactly up to motions.

Anahtar Kelimeler

Extended hyperbolic plane, hyperbolic plane of positive curvature, hyperbolic plane, Raisa Orbit, $R$-orbit, inversion, absolute

Teşekkür

Dear Dr. Prof. Kazim ILARSLAN, Please, accept for publication my new manuscript on conics, which are invariant under inversions with respect to the absolute of an extended hyperbolic plane. With best regards, Lyudmila ROMAKINA

Kaynakça

• [1] Busemann, H., Kelly P.: Projective Geometry and Projective Metrics. Academic Press Inc. New York (1953).
• [2] Glaeser G., Stachel H., Odehnal B.: The Universe of conics. From the ancient Greeks to 21st century developments. Springer Spektrum (2016).
• [3] Halbeisen L., Hungerbühler N.: The exponential pencil of conics. Beitr. Algebra Geom., 59, 549-571 (2018).
• [4] Klein, F.: Vorlesungen über Nicht-Euclidische Geometrie. Verlag von Julius Springer. Berlin (1928).
• [5] Liebmann, H.: Nichteuklidische geometrie. Leipzig (1912).
• [6] Petitjean, S.: Invariant-based characterization of the relative position of two projective conics. In: Emiris I., Sottile F., Theobald T. (eds) Nonlinear Computational Geometry. The IMA Volumes in Mathematics and its Applications. Springer. New York. 151. 189-220 (2009).
• [7] Pevzner, S.: Properties of second-order curves in the Lobatchevsky plane which are dual under the focal directrix. Izv. Vyssh. Uchebn. Zaved. Mat. 5, 39-50 (1961).
• [8] Pevzner, S.: A detailed classification of second-order irreducible curves in the Lobatchevsky plane by means of the focal-director invariants. Izv. Vyssh. Uchebn. Zaved. Mat. 6, 85-90 (1962).
• [9] Rosenfel’d, B.: Noneuclidean spaces. Nauka. Moscow (1969).
• [10] Rosenfel’d, B., Zamakhovskii M.: Geometry of Lie groups. Symmetric, parabolic and periodic spaces. Moscow center for countinuous mathematical education. Moscow (2003).
• [11] Romakina, L.: Oval lines of the hyperbolic plane of positive curvature. Izv. Sarat. Univ. (N. S.), Ser. Mat. Mekh. Inform. 12 (3), 37-44 (2012).
• [12] Romakina, L.: Simple partitions of a hyperbolic plane of positive curvature. Sbornik: Mathematics, 203 (9), 1310-1341 (2012). Translated from Matematicheskii Sbornik 203 (9), 83-116 (2012).
• [13] Romakina, L.: Geometry of the hyperbolic plane of positive curvature. P. 1: Trigonometry. Publishing house of the Saratov university. Saratov (2013).
• [14] Romakina, L.: Geometry of the hyperbolic plane of positive curvature. P. 2: Transformations and simple partitions. Publishing house of the Saratov university. Saratov (2013).
• [15] Romakina, L.: Cycles on the hyperbolic plane of positive curvature. J. Mat. Sciences. 212 (5), 605-621 (2016). Translated from Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 415, 137–162 (2013).
• [16] Romakina, L.: Inversion with respect to a hypercycle of a hyperbolic plane of positive curvature. Journal of Geometry. 107 (1), 137-149 (2016).
• [17] Romakina, L.: Svetlana Ribbons with intersecting axes in a hyperbolic plane of positive curvature. Journal for Geometry and Graphics, 20 (2), 209-224 (2016).
• [18] Romakina, L.: Inversion with respect to a horocycle of a hyperbolic plane of positive curvature. Tr. Inst. Mat. 25 (2), 82-92 (2017).
• [19] Romakina, L.: Inversion with respect to the absolute of an extended hyperbolic plane. In: Proceedings of the International Forum of Mathematical Education dedicated to the 225th anniversary of N. I. Lobachevskii, Oct 18-22/2017, Kazan, RUSSIA. Publishing House of Kazan University. 111-114 (2017).
• [20] Romakina, L.: Elliptic R-orbits of the second order in an extended hyperbolic plane. In: Effective researches of modernity, Scientific articles collection of the 10th Internat. Scientific Conference of Eurasian Scientific Association, Moscow, Oct. 2018, 10, 18–21 (2018).
• [21] Romakina, L.: Inversion with respect to an elliptic cycle of a hyperbolic plane of positive curvature. Tr. Inst. Mat. 27 (1-2), 60-78 (2019).
• [22] Romakina, L.: Construction of cubic curves with a node. Beitr. Algebra Geom., 60 (4), 761-781 (2019).