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 through a point not on line ; however, in the plane, two parallels may be closer to than all others (one in each direction of ).
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.
Suppose that the lines carrying distinct parallel rays met. By definition the cannot meet on the side of which either is on. Then they must meet on the side of opposite to , call this point . Thus . Contradiction.