{{short description|Geometrical term}} {{Cleanup rewrite|date=May 2026}} frame|right|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 <math>R</math> through a point <math>P</math> not on line <math>R</math>; however, in the plane, two parallels may be closer to <math>R</math> than all others (one in each direction of <math>R</math>).
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 {{langx|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== thumb|The ray ''Aa'' is a limiting parallel to ''Bb'', written: <math>Aa|||Bb</math>
A ray <math>Aa</math> is a limiting parallel to a ray <math>Bb</math> if they are coterminal or if they lie on distinct lines not equal to the line <math>AB</math>, they do not meet, and every ray in the interior of the angle <math>BAa</math> meets the ray <math>Bb</math>.<ref>{{cite book|last=Hartshorne|first=Robin|authorlink=Robin Hartshorne|title=Geometry: Euclid and beyond|year=2000|publisher=Springer|location=New York, NY [u.a.]|isbn=978-0-387-98650-0|edition=Corr. 2nd print.}}</ref>
==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 <math>AB</math> which either <math>a</math> is on. Then they must meet on the side of <math>AB</math> opposite to <math>a</math>, call this point <math>C</math>. Thus <math> \angle CAB + \angle CBA < 2 \text{ right angles} \Rightarrow \angle aAB + \angle bBA > 2 \text{ right angles} </math>. Contradiction.
== See also== * horocycle, In Hyperbolic geometry a curve whose normals are limiting parallels * angle of parallelism
==References== <references />
Category:Non-Euclidean geometry Category:Hyperbolic geometry