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

In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line R {\displaystyle R} through a point P {\displaystyle P} not on line R {\displaystyle R}; however, in the plane, two parallels may be closer to R {\displaystyle R} than all others (one in each direction of R {\displaystyle 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: A a | | | B b {\displaystyle Aa|||Bb}

A ray A a {\displaystyle Aa} is a limiting parallel to a ray B b {\displaystyle Bb} if they are coterminal or if they lie on distinct lines not equal to the line A B {\displaystyle AB}, they do not meet, and every ray in the interior of the angle B A a {\displaystyle BAa} meets the ray B b {\displaystyle 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 A B {\displaystyle AB} which either a {\displaystyle a} is on. Then they must meet on the side of A B {\displaystyle AB} opposite to a {\displaystyle a}, call this point C {\displaystyle C}. Thus ∠ C A B + ∠ C B A < 2 right angles ⇒ ∠ a A B + ∠ b B A > 2 right angles {\displaystyle \angle CAB+\angle CBA<2{\text{ right angles}}\Rightarrow \angle aAB+\angle bBA>2{\text{ right angles}}}. Contradiction.

See also