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Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P.

However, there's one line M which is "special" in that the closest that it gets to L is exactly P.

One way to construct this line is to draw the perpendicular from P down to L, then obtain M by drawing the perpendicular through L at P.

Another way, in the Klein model, is to get the unique chord M going through P which is "parallel" to L if L and M are both extended outside of the disk to the rest of the Euclidean plane.

I believe this line should also be the "bisector" of the angle made by the asymptotic limiting parallels to L which go through P.

Does this line have a name in the hyperbolic geometry literature? It is, in a certain sense, the "most parallel" line to L going through P.

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I don't think it has a special name. the line is not so special. (every point has one, two of these lines from different points but the same axis can intersect, while if they are perpendiculatr to the same line they don't)

Maybe best is to describe it as:

"the line trough $P$ that is perpendicular to the perpendicular from $P$ to $l$"

or

"the line trough $P$ that is member of the hyperpencil/ bundle with as centre the perpendicular from $P$ to $l$"

Be Aware:

In hyperbolic geometry there is (yet) no general accepted terminology, different authors use different names for the same "thing" and the same name for different "things". So read and formulate carefully.

As example for some authors an "Ideal point" is a point at the infinite (Greenberg) for some it are the points beyond that ( Sommerville)

In the other direction:

Non-intersecting lines with a common perpendicular are called "ultraparallel lines" (Greenberg) or "hyperparallel lines" (Borsuk)

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