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Hyperbolic Geometry - Lines and Distances

Andrew G. Bennett


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.

The applet

To help get you familiar with hyperbolic lines and distances, I've prepared an applet to let you experiment with them. You will need a Java-enabled browser to run the applet (Netscape 3.0 or higher or Internet Explorer 3.0 or higher on either Windows 95 or a Mac). Once you've read the following instructions, click the link below to launch the applet in a new window. If the resolution of your monitor is 640x480, you will probably want to maximize the window. If the resolution of your monitor is 800x600 or larger, you should see everything just fine the way it comes up on its own.

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

  • Check that if you leave the red line fixed and one blue point fixed, there really are infinitely many lines through the fixed blue point that are parallel to the red line.
  • Observe that distances are larger as you go down toward the edge than as you go up away from the edge. Since the points are shown with width and height (so you can click on them with your mouse), you can't push them all the way to the edge, so you can't actually push them infinitely far apart. (The point represented by each square is at the center of the square drawn on the screen.)
  • Push the two squares as far apart as they will go on the screen, as measured by hyperbolic distance. Where are they farthest apart? (Hint: the farthest apart you can push them is over 90 units apart).

Click here to launch applet. (It will open a new window.)

Andrew G. Bennett, "Hyperbolic Geometry - Lines and Distances," Convergence (January 2005)