I am quite new to hyperbolic geometry so even an answer that this question doesn't make any sense can be very helpful.
As far as i understand:
There are different models of a plane where hyperbolic geometry hold.
lets call this a hyperplane
1 the Beltrami disk model http://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model - hyperbolic lines stay straight lines but all the rest need to be calculated
2 the Poincare disk model,
http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model
- hyperbolic lines become circle arcs
the angles in the poincare disk model are conform the angles in a hyperplane so can be measured
but length of segments still need to be calculated
3 Hyperboloid_model
http://en.wikipedia.org/wiki/Hyperboloid_model
not sure where is it good for? was thinking that it was length conforming,(so length can be measured instead of calculated) but should it than not have saddlepoints (http://en.wikipedia.org/wiki/Saddle_point , because hyperbolic circles are longer than normal flatland circles), and it doesn't have a saddlepoint
so what is the use of the model?