Inversion with respect to a horocycle of a hyperbolic plane of positive curvature

L. N. Romakina

Trudy Instituta matematiki, Tome 25 (2017) no. 1, pp. 82-92.

Résumé

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 single-branch hyperbolic parabola of the plane $\widehat {H}$ if $l$ crosses the horizon of inversion base in two real points.

Détail

Keywords: hyperbolic plane of positive curvature, horizon of the horocycle, inversion with respect to a horocycle of the hyperbolic plane of positive curvature.
Mots-clés : horocycle

Bibliographie

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