Limiting parallel

In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line ${\displaystyle l}$ through a point ${\displaystyle P}$ not on line ${\displaystyle R}$; however, in the plane, two parallels may be closer to ${\displaystyle l}$ than all others (one in each direction of ${\displaystyle R}$).

Thus it is useful to make a new definition concerning parallels in neutral geometry. If there are closest parallels to a given line they are known as the limiting parallel, asymptotic parallel or horoparallel (horo from Greek: ὅριον — border).

For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal.

If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle.