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 — 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 $\angle CAB + \angle CBA < 2 \text \Rightarrow \angle aAB + \angle bBA > 2 \text{ right angles}$. Contradiction.

See also

* horocycle, In Hyperbolic geometry a curve whose normals are limiting parallels
* angle of parallelism

References

Non-Euclidean geometry
Hyperbolic geometry