Limiting parallel



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

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 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.

Definition


A ray $$Aa$$ is a limiting parallel to a ray $$Bb$$ if they are coterminal or if they lie on distinct lines not equal to the line $$AB$$, they do not meet, and every ray in the interior of the angle $$BAa$$ meets the ray $$Bb$$.

Properties
Distinct lines carrying limiting parallel rays do not meet.

Proof
Suppose that the lines carrying distinct parallel rays met. By definition they cannot meet on the side of $$AB$$ which either $$a$$ is on. Then they must meet on the side of $$AB$$ opposite to $$a$$, call this point $$C$$. Thus $$ \angle CAB + \angle CBA < 2 \text{ right angles} \Rightarrow \angle aAB + \angle bBA > 2 \text{ right angles} $$. Contradiction.