Is it possible for a geodesic to intersect two distinct geodesics in the hyperbolic plane?

Question

Considering the hyperbolic plane, is it possible for a geodesic to intersect orthogonally two other, distinct geodesics $L, L'$ that do not intersect with each other at $\infty?$ (that is, $L,L'$ do not intersect with each other at infinity).

Answer 1

In the Poincaré disk model, let $L_0$ be the hyperbolic set corresponding to the Euclidean line $y=0$, and let $L$ and $L'$ be the hyperbolic sets corresponding to the Euclidean circles $(x-\sec \theta)^2+y^2=\tan^2 \theta$, $(x+\sec \theta)^2+y^2=\tan^2 \theta$ for some $\theta \in (0,\pi/2)$. Then $L$ has a radius which is tangent to the unit circle at the point $(\cos \theta,\sin\theta)$, so $L$ is a hyperbolic line, and similarly with $L'$. These lines do not intersect, even at infinity, because they are separated by the line $x=0$. But $L_0$ is perpendicular to both.

Here's a GeoGebra link which hopefully helps illustrate all of this.

(In general, if $L,L'$ are two any non-intersecting geodesics, there will be points $p\in L$ and $q \in L'$ at which that the distance between $L$ and $L'$ is minimized. The line $pq$ is then orthogonal to both $L$ and $L'$.)

Answer 2

In fact this always happens.

For any geodesic $M$, and for any distinct points $p \ne p' \in M$, if we let $L,L'$ denote the unique geodesics that meet $M$ orthogonally at the points $p,p'$ respectively, then the geodesics $L$ and $L'$ do not intersect with each other at infinity (nor finitely for that matter).

One way to see this is to work in the upper half plane model, where $M$ is the geodesic $x=0$. Each of $L,L'$ is a Euclidean semicircle hitting $x=0$ orthogonally, the first at $p = (0,y)$ and the second at $p'=(0,y')$, where $y,y' > 0$ and $y \ne y'$ by assumption. The Euclidean radii of these circles are $y,y'$. The points at infinity of $L$ are $(y,0)$, $(-y,0)$ and the points at infinity of $L'$ are $(y',0),(-y',0)$.