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Journals / International Electronic Journal of Geometry / 2021 / Cilt: 14 - Sayı: 1

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

Lyudmila N. ROMAKINA
Pages
196–206
Publication Date
2022-10-18
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

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.

📌 3 Key Takeaways

🎯
Objective & Hypothesis: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 $\rho$, $\rho \in \mathbb R_+$.
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Key Findings & Data:We prove that each hyperbolic Raisa Orbits of the second order in $H^2$ belongs to one of four conics types of this plane.
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Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Extended hyperbolic plane · hyperbolic plane of positive curvature · hyperbolic plane

Abstract

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.

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Cite This Article

ROMAKINA, L. N. (2022). Hyperbolic Raisa Orbits of the Second Order in an Extended Hyperbolic Plane. International Electronic Journal of Geometry, Cilt: 14 - Sayı: 1, 196–206. https://ajindex.com/articles/539580

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