In Hyperbolic geometry:
What is the relation between the length of the arc of a horocycle between two points and the length of the chord (segment) between the two points?
Also what is the relation between the length of the arc of a horocycle between two points and the maximum distance between the chord (segment) and the horocycle?
Also what is the relation between the length of the arc of a horocycle between two points and the angle between the chord (segment) and the radius of the horocycle?
In short I am just looking for some formula's that can tell me more about arcs of horocycles.
references to formularia very welcome
Here are some hints about how to do this constructively. Suppose $p,q$ are points in the hyperbolic plane such that the length of the arc of the horocycle between $p,q$ is equal to $\ell$. Then we may assume that, in the upper half plane, $p=(0,1)$ and $q=(\ell,1)$.
To get a formula for the length of the geodesic segment between $p,q$, construct a semicircle through $p,q$ hitting the real line at right angles, and integrate the arc length element $\frac{ds}{y}$ along $C$ from $p$ to $q$.
To compute the maximum distance: integrate the arc length element along the vertical segment from $(\ell/2,1)$ up to the point where that vertical segment hits $C$.
And the angle you ask for is just the ordinary Euclidean angle between the circle $C$ and the line $y=1$ at the point $p$.