I was reading somewhere (sorry I always forget where) that the scale of the Poincare Half plane is y (the vertical)
So at the boundary line the scale is $ 0 $ or $ ( 1 : \infty ) $.
at the horocycle $ y = 1 $ the scale is $1:1$ and
at $ y = \infty $ the scale is $ \infty$ and all vertical lines and hypercycles meet .
now I was wondering what is the scale function for the other models of the hyperbolic plane, escpecially for the Poincare disk model and the Klein disk model?