Hyperbolic Geometry

Triangles, Angles, and Area

The Poincare half-plane model is conformal, which means that hyperbolic angles in the Poincare half-plane model are exactly the same as the Euclidean angles (with the angles between two intersecting circles being the angle between their tangent lines at the point of intersection. A hyperbolic triangle is just three points connected by (hyperbolic) line segments. Despite all these connetions, hyperbolic triangles are quite different from Euclidean triangles.

Since the hyperbolic line segments are (usually) curved, the angles of a hyperbolic triangle add up to strictly less than 180 degrees. Since a quadrilateral can always be cut into two triangles, a quadrilateral must have its angles sum to less than 360 degrees, so in hyperbolic geometry there are no squares, which makes defining area in terms of square units difficult. Defining area in hyperbolic geometry is quite technical and we won't worry much about it in this class (stop by and see me if you want to go over the details, but be sure you have a fair bit of time to spend :-)

Two other major differences between hyperbolic triangles and Euclidean triangles are

While the sides of hyperbolic triangles can get as large as you want, the area of any triangle is less than pi.
There is no concept of similar triangles, if two triangles have the same angles then they are congruent (though they may look different in the model since lengths appear different depending on how close they are to the edge, and that goes for area too).

The applet

To help get you familiar with hyperbolic triangles, 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 work). Once you've read the following instructions, click the link below to launch the applet. It will launch in a new window. If the resolution of your monitor is 640x480, you will probably be best off maximizing 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 a red point, a blue point, and a green point. To help the green pont show up well, the background is now a light gray, though the edge of the half-plane is still marked with the same (darker) gray as in previous applets. The points are connected by (hyperbolic) line segments to make a triangle. The angles of the triangle are listed below the edge of the half-plane. They are color coded, so the angle at the red point is given in red, etc. Below these angles are the sum of the three angles and the area of the triangle. Clickyour mouse on a point and drag it (while holding the mouse button down) to move the point. The triangle will follow the point as in the other applets. (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 again.)

Things to try

Move the points around to check how the angle sums behave. Observe that the smaller the traingle, the closer the sum of the angles is to 180 degrees.
What is the biggest area you can make in this applet? Since the points are shown with width and height so you can click on them with your mouse, you can't push them out to infinity, but you can get them far enough apart to get an area of over 3. Where do you move the points to get the biggest area?
What is the smallest angle sum you can make in this model. As above, you won't be able to reach the theoretical minimum, but you can get reasonably close.

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

Please report any problems with this web site to bennett@math.ksu.edu.