Go to the webpage for the non-Euclid applet. There is a link from the math 342 home page to this page, which is found at http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html.
This applet provides a dynamic geometry model of one kind of hyperbolic geometry, called the Poincaré Disk model. Notice that it is a kind of taxicab geometry: the distance between two points in the Poincaré disk is given by a non-Euclidean distance function.
Experiment with this model. It seems very different from the hyperbolic planes that we have been looking at. Do you think it is reasonable to call this a model of hyperbolic geometry? Why or why not? You may also find it useful to compare this model with the other model provided by non-Euclid, which is called the Upper Half-Plane model.
Try to come up with as many examples as you can of theorems of Euclidean geometry that are not true in the Poincaré Disk model, and of facts that appear to be true in the Poincaré Disk model that are not true in Euclidean geometry.