Hyperbolic geometry is a geometry for which we accept the first four axioms of Euclidean geometry but negate the fifth postulate, i.e., we assume that there exists a line and a point not on the line with at least two parallels to the given line passing through the given point. This corresponds to doing geometry on a surface of constant negative curvature. Such a geometry is very different from the familiar Euclidean geometry. This module has three applets designed to get you familiar with some of the basic properties of the hyperbolic plane.
Unfortunately, the hyperbolic plane can't be embedded in Euclidean 3-space. In order to represent the hyperbolic plane on a computer monitor, we must flatten out the curvature. In so doing, many of the straight lines in hyperbolic space become curved. One of the standard models of flattening out the hyperbolic plane is due to the French mathematician Henri Poincaré. In this model, the hyperbolic plane is squashed onto a Euclidean half-plane. The following links will take you to discussions of different features of this model. The discussions will make more sense if you view them in order.