I'm asking a question about a construction due to Thurston.
Let's consider a hyperbolic triangle (I'm considering the Poincarè disc model of the hyperbolic plane) and from each one of the three vertices let's foliate the triangle with horocycles untill we reach the central zone bounded by three horocycles. I made a (bad) picture: the geodesics are in red and the horocycles in blu.
Now let's choose one spike on the triangle and the horocycle at hyperbolic distance $t$ from the central, unfoliated zone. My question is:
How do I compute the hyperbolic length of this horocycle? Does it vary linearly with $t$?
I made another picture:
I'm new to hyperbolic geometry so I apoligise if my question is trivial. And I'm sorry if I'm not explaining my attempt to solve the problem, but this seems rather complicated to me, so I really need some help understanding where to start to solve it.