Cutoff

File:Limiting Parallels.jpg

The ray Aa is a limiting parallel to Bb, written: $Aa\left|\right||Bb$

In neutral geometry, there may be many lines parallel to a given line $l$ at a point $P$, however one parallel may be closer to $l$ than all others. Thus it is useful to make a new definition concerning parallels in neutral geometry. If there is a closest parallel to a given line it is known as the limiting parallel. The relation of limiting parallel for rays is an equivalence relation, which includes the equivalence relation of being coterminal.

Limiting parallels may sometimes form two, or three sides of a limit 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 the 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 $\mathrm{\angle }CAB+\mathrm{\angle }CBA<2\text{ right angles}\Rightarrow \mathrm{\angle }aAB+\mathrm{\angle }bBA>2\text{ right angles}$. Contradiction.