# Horocycle

Jump to: navigation, search

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.

How to Cite This Entry:
Horocycle. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Horocycle&oldid=14909
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098