What's the point of the Poincaré disc model? I'm trying to work out the point of the Poincaré disc model (excuse the pun). As far as I can tell, it's a disc, on which the only permitted lines are a line straight across the middle, and arcs of circles that meet the edge of the disc at right angles (and segments of such arcs, so if I draw three of them together I can get a curvy-edged triangle), and hyperbolic circles (which is "just" the set of points that are a common distance from another point, but where distance is a newly defined kind of distance).
We define new meanings for distance and area and such on the disc. So now I've got a disc on which I can draw some curves and say they are of length such-and-such (using my new definition of distance), and I can say a curvy triangle is of area such-and-such, and I can prove that various allowed lines will and will not touch/cross under various circumstances and so on. I've calculated various distances and mid-points and areas and reflections and so on.
But I could come up with an infinite number of geometries by allowing lines of various types and defining length and area in my own special way (or can I?). What's so special about this disc that it's worth examining? I'm not averse to playing with new ideas just for fun, but this Poincaré disc seems like a big deal, so it must be for more than the fun of it.
Perhaps the clue is in the name; it's a Poincaré disc model, so what's it a model of?
I think if I have a hyperbolic space with some geodesics on it, I can project those geodesics onto a disc and the projections are the right shape to be allowed (i.e. arcs of circles meeting the edge of the disc at right-angles), but I don't see what that gets me that I didn't already know. Is there some link between the length of the line on the disc (as defined using my new special definition of length) that relates to the length of the geodesic in the original hyperbolic space? Can I look at the curves on the disc and from them know something about the curves in the hyperbolic space that I didn't know before? Why have I gone to the trouble of creating this disc with a new definition of length etc? 
 A: It's a model of hyperbolic geometry in the plane, the same geometry described by Lobachevsky. There are several models of planar hyperbolic geometry. The most common ones include the Poincaré disk (or more generally ball), the Poincaré upper half-plane (or more generally half-space), the Beltrami-Klein model which is somtimes also called the projective model, and the hyperboloid model which uses a three-dimensional Minkowsky space to embed the plane.
Comparing these models, the Poincaré disk model has the benefit that it doesn't extend to infinity (as the upper half-plane does), and preserves angles (contrary to the non-Poincaré models), and lives in the plane without need for a third dimension (as needed by the hyperboloid). But these benefits are balanced by drawbacks: the upper half-plane can use real $2\times2$ matrices to describe isometric transformations, the Beltrami-Klein model uses straight lines to model geodesics, and the hyperboloid model is very close in formulation to the geometry on a sphere.
An imperfect model of this geometry would – at least in my interpretation of the word “imperfect” – be the geometry on the tractricoid. Since the tractricoid has constant negative curvature, it closely resembles the hyperbolic plane, uisng “real” angles and “real” geodesics, as they are in the ambient three-space. But the tractricoid only models a part of the hyperbolic plane; it contains closed curves which would not close in the hyperbolic plane. So it is only a local model, which doesn't represent the global structure well. There can be no embedding of the hyperbolic plane into real three-space which uses normal Euclidean angles and geodesics.
You could define all kinds of stuff about what you call a “line”, what you call “distance” and so on. The special thing about hyperbolic geometry is the fact that it still satisfies the first four axioms of Euclid, even though it violates the fifth. In this regard it is pretty much unique.
