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oricycle, limiting line

The orthogonal trajectory of parallel lines in the Lobachevskii plane in a certain direction. A horocycle can be considered as a circle with centre at infinity. Horocycles generated by one pencil of parallel lines are congruent, concentric (i.e. cut out congruent segments on the lines of the pencil), non-closed, and concave to the side of parallelism of the lines of the pencil. The curvature of a horocycle is constant. In Poincaré's model, a horocycle is a circle touching the absolute from within.

A straight line and a horocycle either do not have common points, touch each other, intersect at two points at the same angle, or intersect at one point at a right angle.

Two, and only two, horocycles pass through two points of the Lobachevskii plane.

Two crossing horocycles in Poincaré disc model


[1] V.F. Kagan, "Foundations of geometry" , 1–2 , Moscow-Leningrad (1949–1956) (In Russian)
[2] A.P. Norden, "Elementare Einführung in die Lobatschewskische Geometrie" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[3] N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
How to Cite This Entry:
Horocycle. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article