Limiting parallel

In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line

; however, in the plane, two parallels may be closer to

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.

is a limiting parallel to a ray

if they are coterminal or if they lie on distinct lines not equal to the line

, they do not meet, and every ray in the interior of the angle

[1] Distinct lines carrying limiting parallel rays do not meet.

Suppose that the lines carrying distinct parallel rays met.

By definition they cannot meet on the side of