Why to include the $C$ in the formula for the distance in hyperbolic geometry?

Question

I'm reading Penrose's: Road To Reality. First he gives the Lambert formula and later, he says that if you want, you can include the $C$ of the Lambert area formula. But It's not clear why I would include that, from his presentation, it seems that the $C$ has to do with scaling but It's not clear what's the role of the scaling in hyperbolic geometry.

Answer 1

$C$ is related to the curvature. If the model being used is the upper half plane, the curvature is $-1$ and $C=1.$ A book that gives full detail on other choices is George Martin, The Foundations of Geometry and the Non-Euclidean Plane. The quantity that Martin calls the "distance scale" $k$ is defined in terms of a relationship among horocycles...

It may help to consider trigonometry on the sphere, then vary the radius of the sphere. Very similar, but instead of the angle sum being below $\pi,$ on the sphere it is always above $\pi.$