Problem in understanding models of hyperbolic geometry I recently started reading The Princeton Companion to Mathematics. I am currently stuck in the introduction to hyperbolic geometry and have some doubts about its models.
Isn't the hyperbolic space produced by rotating a hyperbola? That is, isn't hyperbolic geometry carried out on a hyperboloid?
Also, how is the half plane model generated? Is it generated by taking the crossection of the hyperboloid on the upper half of the complex plane or by taking a projection?
Also, can someone explain to me how "hyperbolic distances become larger and larger, relative to Euclidean ones, the closer you get to the real axis". Is the axis referred to here the real axis of the complex plane or some other axis. And why do distances become larger the closer we get to the real axis? The part of the line close to the real axis simply looks like a part of a circle whose distance does not increase abnormally.
Any visuals will be appreciated.

It looks like you are talking about the Poincaré half-plane model of hyperbolic geometry (see https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model )
Be aware this is only a model of hyperbolic geometry.
For doing "real-life" hyperbolic geometry you need a "surface with a constant negative curvature " or pseudo spherical surface see https://en.wikipedia.org/wiki/Pseudosphere
Also the hyperboloid is only a model of hyperbolic geometry  (the curvature of an hyperboloid is positive and not constant, so wrong on both counts of the curvature ) 
Having said all this you seem to investigate the Poincaré half-plane model of hyperbolic geometry and indeed the scale of this model becomes 0 when you get near the x=0 line
also see 
https://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane
for more information 
Hopes this helps
