Common perpendicular of ultraparallel hyperbolic lines

Two disjoint hyperbolic lines with four distinct ideal endpoints have a unique common perpendicular, and the segment of that perpendicular between them realizes their minimum distance. An upper-half-plane isometry can make the two lines concentric semicircles; their common perpendicular then becomes the imaginary axis.

Three pairwise ultraparallel hyperbolic lines may have disjoint common perpendiculars or coincident ones. In the upper-half-plane model, three semicircles of radius $1/2$ centred at $-5/2,0,5/2$ have mutually disjoint common perpendiculars. Three concentric semicircles instead have the same vertical common perpendicular. Thus pairwise ultraparallelism alone does not determine the incidence of their perpendiculars.