Hyperbolic line

A hyperbolic line is a complete geodesic in the hyperbolic plane. In the Poincare half-plane model, hyperbolic lines are vertical rays and semicircles orthogonal to the boundary.

An ideal endpoint of a hyperbolic line is one of its two limiting points on the boundary at infinity. In the Poincare disc model these lie on the unit circle; in the upper half-plane model they lie on $\mathbb{R}\cup \{\infty \}$. These are boundary points, not points of the hyperbolic plane. Any two distinct ideal endpoints determine a unique hyperbolic line.

Two distinct hyperbolic lines are parallel, in the limiting-parallel convention, when they do not intersect in the hyperbolic plane and have exactly one common ideal endpoint. In the upper half-plane model, distinct vertical lines are parallel, sharing infinity. Disjoint lines with no common ideal endpoint are instead ultraparallel hyperbolic lines; only these have a common perpendicular of ultraparallel hyperbolic lines.

Two hyperbolic lines are ultraparallel when they are disjoint and have no common ideal endpoint. They possess a unique common perpendicular, which realizes the distance between them.