Horocycle

In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature which converges asymptotically in both directions to a single ideal point, called the centre of the horocycle. The perpendicular geodesics through every point on a horocycle are limiting parallels, and also all converge asymptotically to the centre. It is the two-dimensional case of a horosphere.

In Euclidean space, all curves of constant curvature are either straight lines (geodesics) or circles, but in a hyperbolic space of sectional curvature ${\displaystyle -1,}$ the curves of constant curvature come in four types: geodesics with curvature ${\displaystyle \kappa =0,}$ hypercycles with curvature ${\displaystyle 0<|\kappa |<1,}$ horocycles with curvature ${\displaystyle |\kappa |=1,}$ and circles with curvature ${\displaystyle |\kappa |>1.}$

Any two horocycles are congruent, and can be superimposed by an isometry (translation and rotation) of the hyperbolic plane.

A horocycle can also be described as the limit of the circles that share a tangent at a given point, as their radii tend to infinity, or as the limit of hypercycles tangent at the point as the distances from their axes tends to infinity.

Two horocycles with the same centre are called concentric. As for conceptric circles, any geodesic perpendicular to a horocycle is also perpendicular to every concentric horocycle.