I am comfortable using the Poincare disk as a model of hyperbolic geometry, but it has left me with a question for which I don't have a good answer:
Is there a negatively curved surface that we could imagine the Poincare disk as representing? Ideally there would be a reasonably nice way to transform this negatively curved surface into the Poincare disk (similar to how the hyperboloid model can be turned into the Poincare disk).
Note: When I say "negatively curved surface," I am looking for a surface embedded in $\mathbb{R}^3$ whose metric induced by the Euclidean metric causes it to exhibit hyperbolic geometry, just as the sphere embedded in $\mathbb{R}^3$ exhibits elliptic geometry. (I'm under the impression that this will require the desired surface to have constant curvature, but I'm not sure about this.)