Alternative definition on parallel lines

Question

I was puzzling about an alternative definition of a parallel line . With parallel lines is I mean converging (horo) parallel line .

I wanted a definition that did not mention ideal points or lines intersecting at the absolute or anything like that.

And after some puzzling I came to the following idea:

The two lines a and b are horoparallel if and only if for every point A on a, if C is the point on b nearest to A then there is a point D on a that is nearer to C than A.

Is this a definition correct or did I overlook something?

Answer 1

The definition of parallel lines in hyperbolic geometry is that $l$ and $m$ are parallel if they do not intersect and further for each point $P$ on $l$ any line though $P$ lying between $l$ and $m$ will intersect $m$.