View source for Horocycle

''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.

[[File:Horocycles.svg|center|200px|Two crossing horocycles in Poincaré disc model]]

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,   "Foundations of geometry" , '''1–2''' , Moscow-Leningrad  (1949–1956)  (In Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Norden,   "Elementare Einführung in die Lobatschewskische Geometrie" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.V. Efimov,   "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian) {{ZBL|0108.16203}}</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1961)</TD></TR>
</table>