Hyperbolic Models

There are other models of hyperbolic geometry besides the saddle. The problem with the saddle model is that it seems to flatten out as the edges go to infinity. Actually, a hyperbolic surface acts as if every point is the saddle center.

horn

 The pseudosphere is a trumpet-shaped surface that extends infinitely at both the pointed end and the bell edge. The flare of the bell shows the negative curvature of the surface. This model is difficult to imagine going to infinity.
disk
 The Poincare disk is named after a French mathematician. He designed this hyperbolic model as the interior of a circle -- without any "edge." The dotted line on the model shows that the surface goes to infinity in all directions. Imagine looking straight down the "throat" of the pseudosphere.
crochet
 A hands-on model can be made by crocheting a surface that becomes more "curly" as it approaches the infinite edge. Choose two points on the edge of this model and find the shortest path between them. Unless the points are straight across from each other, the path you choose will arc into the center of the surface, away from the ruffles.