How to graph in hyperbolic geometry?

Question

I was given the following question regarding hyperbolic geometry:

In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line $x=7$.

How do I go about doing this? I am very new to non-Euclidean geometry. Thank you.

Answer 1

Hyperbolic geometry distinguishes different concepts of parallelity. So here are the terms I'm familiar with:

• Two lines are parallel if they don't share a point of the hyperbolic plane.
• Two lines are limiting parallel if they meet at an ideal point.
• Two lines are ultraparallel if they are parallel but not limiting parallel.

For a line and a point not on that line, there are exactly two limiting parallel lines. The ideal points of the upper half plane model are the points on the horizontal axis, together with the point at infinity. A hyperbolic line would be modeled either by a vertical line (which passes through the point at infinity) or a circle perpendicular to the horizontal axis (i.e. with its center on said axis). So in your case, the ideal points of the line $x=7$ would be the point $(7,0)$ and the point at infinity. Therefore, the limiting parallel lines would be modeled by a vertical line through the point $(3,1)$ and by a circle with center on the horizontal axis which passes through $(3,1)$ and $(7,0)$.

An ultraparallel line would be anything between these two cases.

This figure has the given point and line in red, the limiting parallels in blue and some ultraparallel lines in cyan.