# 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.

#### References

[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) |

#### Comments

#### References

[a1] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) |

**How to Cite This Entry:**

Horocycle. A.B. Ivanov (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Horocycle&oldid=14909