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Hyperbolic Geometry - Triangles, Angles, and Area

Andrew G. Bennett


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 similarities, 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 add up to less than 360 degrees, so in hyperbolic geometry there are no squares, which makes defining area in terms of square units difficult. It turns out, however, that there is a unique function (up to multiplication by a scale factor) that satisfies the usual area axioms:

  • Area is non-negative.
  • If you cut up a figure into a finite number of pieces and rearrange them to form a second figure, then the two figures must have the same area.

For a triangle, this function is the defect, defined as

where the angles alpha, beta, and gamma are measured in radians.

In addition to the smaller angle sum, there are two other major differences between hyperbolic triangles and Euclidean triangles:

  • 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. (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). 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 a red point, a blue point, and a black point. 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. Click your 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 triangle, 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 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. (It will open a new window.)

Andrew G. Bennett, "Hyperbolic Geometry - Triangles, Angles, and Area," Convergence (January 2005)