Limiting parallel
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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.
If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle.
A ray is a limiting parallel to a ray if they are coterminal or if they lie on distinct lines not equal to the line , they do not meet, and every ray in the interior of the angle meets the ray .[1]
