A horocycle in the hyperbolic plane is a curve orthogonal to every geodesic approaching a fixed ideal point. Equivalently, it is the limiting position of a circle in hyperbolic geometry whose center approaches that ideal point while one point of the circle is held fixed.
In the Poincaré disk, a horocycle is represented by a Euclidean circle internally tangent
to the boundary circle. In the upper half-plane realization
of the hyperbolic plane, it is represented by
either a Euclidean circle tangent to the boundary line
or a line parallel to the boundary. For a hyperbolic
plane normalized to have Gaussian curvature 
, every horocycle has constant geodesic
curvature of absolute value 1.
