
Inversion with respect to a horocycle of a hyperbolic plane of positive curvature
L. N. Romakina^{} ^{} Saratov State University
Abstract:
Inversion with respect to a horocycle of the hyperbolic plane $\widehat{H}$ of positive curvature in Cayley – Klein projective model is investigated. Analytical expression of inversion in the canonical frame of the second type is received. Images of the lines and oricycles, concentric with base of inversion are defined. The image of the line $l$ of the plane $\widehat{H}$ which isn't containing the inversion center is: 1) a parabola of the Lobachevskii plane if $l$ has no common real points with the horizon of inversion base; 2) an equidistant line of the Lobachevskii plane if $l$ concerns the horizon of inversion base; 3) a singlebranch hyperbolic parabola of the plane $\widehat {H}$ if $l$ crosses the horizon of inversion base in two real points.
Keywords:
hyperbolic plane of positive curvature, horocycle, horizon of the horocycle, inversion with respect to a horocycle of the hyperbolic plane of positive curvature.
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UDC:
514.133 Received: 21.01.2017
Citation:
L. N. Romakina, “Inversion with respect to a horocycle of a hyperbolic plane of positive curvature”, Tr. Inst. Mat., 25:1 (2017), 82–92
Citation in format AMSBIB
\Bibitem{Rom17}
\by L.~N.~Romakina
\paper Inversion with respect to a horocycle of a hyperbolic plane of positive curvature
\jour Tr. Inst. Mat.
\yr 2017
\vol 25
\issue 1
\pages 8292
\mathnet{http://mi.mathnet.ru/timb270}
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http://mi.mathnet.ru/eng/timb270 http://mi.mathnet.ru/eng/timb/v25/i1/p82
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