In the Poincaré half-plane model, the hyperbolic plane is flattened into a Euclidean half-plane. As part of the flattening, many of the lines in the hyperbolic plane appear curved in the model. Lines in the hyperbolic plane will appear either as lines perpendicular to the edge of the half-plane or as circles whose centers lie on the edge of the half-plane. Note that the edge of the half-plane itself (marked in gray in the picture) is not part of the hyperbolic plane.
With these definitions it is not hard to show that two points determine a line, as is required by Euclid's Axiom 1. It may initially appear that the second axiom, that any segment can be extended indefinitely, is violated by the existence of the edge. The trick is that distance is defined so that the edge is infinitely far away. To review the definition of distance, click here.
In the applet you will have two red points and two blue points, with each pair of points defining a hyperbolic line. Click your mouse on a point and drag it (while holding the mouse button down) to move the point. The line will follow the point. Off the edge of the half-plane (marked in gray), you will see the hyperbolic distances between the red points and between the blue points. You will also see a note about whether the lines are parallel. (Bug warning: Sometimes when the window is covered and then uncovered by other windows on your computer monitor the applet doesn't redraw itself completely. If you click on a point and move it, or if you minimize and then restore the window, the applet will redraw itself properly.)
Things to try
Click here to launch applet. (It will open a new window.)