# Definition of the hyperbolic plane

I have found it next to impossible to find a definition of the hyperbolic plane. I am aware of all sorts of imperfect models of the hyperbolic plane -- the pseudosphere, the Poincare disc.

I also know several of its characteristics -- multiple "straight" lines through a given point not meeting another straight line not containing the point -- a surface in space of everywhere negative curvature satisfies the above hyperbolic axiom, etc. etc.

I would like a definition of just what the pseudosphere and the Poincare disc are imperfect models of. Wiki, Needham, Climenhaga etc. etc. do not provide explicit definitions. This is very unusual for a mathematical object of such interest.

• I don't know why you say that the models are “imperfect”, but if you prefer an axiomatic definition, have a look at Robin Hartshorne's Geometry: Euclid and Beyond, section 40. – Hans Lundmark Oct 13 '17 at 15:15
• Bonola has translations of both Bolyai and Lobachevski store.doverpublications.com/0486600270.html – Will Jagy Oct 13 '17 at 16:39
• A closed loop in the plane can be shrunk to a point, but not all loops on the pseudosphere can be. Also lines can't be extended in all directions on the pseudosphere -- but the reference you gave was quite helpful -- thanks. Also Hilbert showed that even though any plane is two dimensional, 3 dimensional Euclidean geometry can't accommodate a model of the hyperbolic plane (Needham p. 326). – luysii Oct 13 '17 at 18:06
• Your question reminds me of What's the point of the Poincaré disc model? Both you and that author seem to associate the term “model” with the engineering sense of “imperfect approximation” instead of the definition in model theory as a structure satifying a given set of formulas like the axioms of hyperbolic geometry. – MvG Oct 15 '17 at 15:57
• Regarding the term pseudosphere, note that different people use it to denote different things. While some use it as a synonym for the tractricoid, others use it as a generic term for any hyperbolic surface. See this anser of mine arguing that this is the meaning Beltrami used, for example. Depending on which definition of the term you use, the model may be imperfect or not. – MvG Oct 15 '17 at 16:04