A horocycle is an curve infinite in length within a hyperbolic plane such that any two non-coincident lines that are perpendicular bisectors of the curve are limiting parallel lines. The centre of a horocycle is the ideal point that these lines asymptotically converge to. Any two horocycles are congruent. A horocycle is a one-dimensional horohypersphere.
A horcycle is the limit of circles or hypercycles that share a tangent line passing through a given point as their radii tend to infinity. Being a one-dimensional horohyperball, tilings of a one-dimensional Euclidean space (Euclidean line) embedded in a hyperbolic space will be circumscribed by a horocycle. The only uniform tiling of the Euclidean line is the apeirogon, which can be constructed by partitioning a Euclidean line into equal length line segments and defining one half-plane as the interior. Apeirogons in a hyperbolic plane can be constructed by partitioning a horocycle into equal length arcs, considering the union of all the line segments that connect the endpoints of each arc as the boundary, and the union of all the omega triangles defined by the ideal point and the endpoints of an arc as the interior.
In the Poincaré disk, horocycles are represented by circles internally tangent to the boundary circle. The point of tangency of the circle is the ideal point that is the centre of the horocycle.