Suppose we are looking at the hyperbolic plane $\mathbb{H}$ with usual metric. Now let $u,v \in \mathbb{R} (u < v)$ and consider the unique geodesic joining them. Now consider horocylces at both $u and v$ with radius $h$ and $k$ respectively. I want to find the distance between the horocycles along the geodesic joining $u$ and $v$.
The standard idea I have to attack this problem is to map the geodesic to the imaginary axis where it is much easier to calculate distance. The mobius map I would use is $z \mapsto \frac{z-v}{z-u}$. However, I can't work out what happens to the horocycles.
By definition both horocycles meet parallel to the real axis, and since mobius maps preserve angles and map circles and lines to circles and lines, then we gain some insight to where the horocycles get mapped to. But this is where I get confused!
Since angles are preserved and circles\lines get mapped to circles\lines then the horocyle at $v$ gets mapped to a circle\line parallel to $\mathbb{R}$ which touches $0 \in \mathbb{C}$. But the imaginary axis will therefore cut this circle at $0$ in half. And since angles are preserved this means that the geodesic through $u$ and $v$ cuts the horocycle at $v$ in half too. Which is clearly false!
I'm not sure what's going wrong here! Any help would be much appreciated!
You're using your Euclidean intuition of distance to reach a clearly not clearly false conclusion! That angles are preserved only means that the geodesic between $u$ and $v$ is perpendicular to any horocycle tangent to $u$ or $v$. It doesn't mean that being "cut in half" is preserved, and in fact, there's not even an unambiguous notion of "half" for a horocycle, which has infinite length.
As you have correctly shown, a horocycle tangent to $0$ is a circle in $\mathbb{H}$ tangent to $0$, and it is perpendicular to the imaginary axis. Horocycles tangent to $\infty$ are horizontal lines parallel to $\mathbb{R}$. To find the distance between the two horocycles, simply integrate up the imaginary axis between their heights.