Limiting parallel explained

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 el|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

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 .^[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

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 rightangles ⇒ \angle aAB + \angle bBA > 2 rightangles . Contradiction.

See also

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

References

1. Book: Hartshorne, Robin. Robin Hartshorne

   . Robin Hartshorne. Geometry: Euclid and beyond. 2000. Springer. New York, NY [u.a.]. 978-0-387-98650-0. Corr. 2nd print..