I have a question about the tesselation of the upper half plane via Ford Circles. Wikipedia says
By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the Poincaré half-plane model) Ford circles can also be interpreted as a tiling of the hyperbolic plane by horocycles.
As far as I understand, the tiling is done by the hyperbolic triangles we get from the Ford Circles and not by the circles itself, is that right? And why do I need horocycles here? Can't I just say that the tiling is done by triangles, whose corners lie on the boundary $\partial \mathbb{H}$ (i.e., are ideal points)?
I think there are mistakes in the wikipedia article
It is not about horocycles but about apeirogons ( https://en.wikipedia.org/wiki/Apeirogon#Apeirogons_in_hyperbolic_plane ,infinite sided polygons) and it is not about the geodesics that intersect where the horocycles meet but about the geodesic that is tangent where the horocycles meet. (the formula given for the geodesic with intersection points is not the geodesic that you need).
From this tangent geodesic take the segment that contains the tangent point and that is between the points where they meet other geodesics
the polygon by connecting these tangent segments form an apeirogon.
in hyperbolic geometry apeirogons have a side length s that depends on the angle between two consequtive sides
$$\Pi(\frac{1}{2} \alpha) = 2s $$
($\Pi$ is the angle of parallelism function, https://en.wikipedia.org/wiki/Angle_of_parallelism )
In the particular Ford circle case:
the apeirogon is the circumscribed apreigon of the horocycle represented by the Ford circle (so the Ford circle is the inscribed horocycle of the apeirogons)
Hopes this helps