![]() Where are they farthest apart? (Hint: the farthest apart you can push them is over 90 units apart).Ĭlick here to launch applet. Push the two squares as far apart as they will go on the screen, as measured by hyperbolic distance.(The point represented by each square is at the center of the square drawn on the screen.) 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. Observe that distances are larger as you go down toward the edge than as you go up away from the edge.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.If you click on a point and move it, or if you minimize and then restore the window, the applet will redraw itself properly.) ( 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. You will also see a note about whether the lines are parallel. 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. ![]() Click your mouse on a point and drag it (while holding the mouse button down) to move the point. In the applet you will have two red points and two blue points, with each pair of points defining a hyperbolic line. If the resolution of your monitor is 800圆00 or larger, you should see everything just fine the way it comes up on its own. If the resolution of your monitor is 640x480, you will probably want to maximize the window. Once you've read the following instructions, click the link below to launch the applet in a new window. 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). To help get you familiar with hyperbolic lines and distances, I've prepared an applet to let you experiment with them. To review the definition of distance, click here. The trick is that distance is defined so that the edge is infinitely far away. It may initially appear that the second axiom, that any segment can be extended indefinitely, is violated by the existence of the edge. With these definitions it is not hard to show that two points determine a line, as is required by Euclid's Axiom 1. Note that the edge of the half-plane itself (marked in gray in the picture) is not part of the hyperbolic plane. 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. As part of the flattening, many of the lines in the hyperbolic plane appear curved in the model. ![]() In the Poincaré half-plane model, the hyperbolic plane is flattened into a Euclidean half-plane. The discussions will make more sense if you view them in order. The following links will take you to discussions of different features of this model. In this model, the hyperbolic plane is squashed onto a Euclidean half-plane. ![]() One of the standard models of flattening out the hyperbolic plane is due to the French mathematician Henri Poincaré. In so doing, many of the straight lines in hyperbolic space become curved. In order to represent the hyperbolic plane on a computer monitor, we must flatten out the curvature. Unfortunately, the hyperbolic plane can't be embedded in Euclidean 3-space. This module has three applets designed to get you familiar with some of the basic properties of the hyperbolic plane. ![]() Such a geometry is very different from the familiar Euclidean geometry. This corresponds to doing geometry on a surface of constant negative curvature. 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. He is also an Associate Editor of this journal. Bennett is in the Department of Mathematics at Kansas State University. ![]()
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