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 ${\displaystyle R}$).

The two lines through a given point P and limiting parallel to line 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.

Definition

The ray Aa is a limiting parallel to Bb, written: ${\displaystyle Aa|||Bb}$

A ray ${\displaystyle Aa}$ is a limiting parallel to a ray ${\displaystyle Bb}$ if they are coterminal or if they lie on distinct lines not equal to the line ${\displaystyle AB}$, they do not meet, and every ray in the interior of the angle ${\displaystyle BAa}$ meets the ray ${\displaystyle Bb}$.^[1]

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 ${\displaystyle AB}$ which either ${\displaystyle a}$ is on. Then they must meet on the side of ${\displaystyle AB}$ opposite to ${\displaystyle a}$, call this point ${\displaystyle C}$. Thus ${\displaystyle \angle CAB+\angle CBA<2{\text{ right angles}}\Rightarrow \angle aAB+\angle bBA>2{\text{ right angles}}}$. Contradiction.

See also

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