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

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International Electronic Journal of Geometry
Volume:14 Issue:1
Hyperbolic Raisa Orbits of the Second Order in an Extended Hyperbolic Plane

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

Pages : 194-206

View : 54 | Download : 11

Publication Date : 2021-04-15

Article Type : Research Paper

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 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(or all elliptic cycles); contains a unique conic defined exactly up to motions.
Keywords : Extended hyperbolic plane, hyperbolic plane of positive curvature, hyperbolic plane, Raisa Orbit, R orbit, inversion, absolute

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