I need to write up a quickie description of Hyperbolic Geometry for non-mathematicians. I am trying to say "Hyperbolic Geoemtry is the Geometry of the surface of a ____"
I remember that there is, in fact, a term for the surface I am thinking of. It is a surface of constant curvature -1, and the model of the Hyperbolic Plane on this surface lends itself very easily to seeing the "skinny triangles" so to speak (ones whose angle measures sum to less than 180). Does anyone remember what the hell this surface is called? I think it starts with a 't'? I honestly can't remember and Google isn't helping.
Thanks in advance, Stackexchange.
In general, a surface of constant negative curvature is called a pseudosphere, being the hyperbolic equivalent of a sphere. A quick search reveals that the particular shape you're likely thinking of, which is the unique solid of revolution of constant negative curvature, is a tractricoid. Though, notably, it is not globally isomorphic to the hyperbolic plane, as it contains a non-smooth "equator".