Horocycle

A horocycle in the Poincare disc model is a Euclidean circle internally tangent to the boundary circle. In the upper half-plane model, a horocycle is a Euclidean circle tangent to the real axis from above, or a horizontal line $y=h>0$. The common geometric feature is a single ideal centre of a horocycle.

Two horocycles have a unique common orthogonal hyperbolic line when their ideal centres of horocycles differ: it is the line with those two ideal endpoints. If the centres agree, an isometry puts both horocycles on horizontal lines; every vertical hyperbolic line meets both orthogonally, so there are infinitely many. This classification remains valid for intersecting or tangent horocycles.

The ideal centre of a horocycle is its point of tangency to the boundary at infinity; a horizontal horocycle in the upper half-plane model has centre infinity. Sending the centre to infinity by an isometry makes the horocycle horizontal. Orthogonal hyperbolic lines then become vertical, proving that a hyperbolic line meets a horocycle orthogonally exactly when it has the centre as an ideal endpoint.