Hyperbolic Geometry - Circles

Author(s):
Andrew G. Bennett

A circle in the hyperbolic plane is the locus of all points a fixed distance from the center, just as in the Euclidean plane. Therefore, the hyperbolic plane still satisfies Euclid's third axiom. A hyperbolic circle turns out to be a Euclidean circle after it is flattened out in the Poincare half-plane model. The only difference is that, since distances are larger nearer to the edge, the center of the hyperbolic circle is not the same as the Euclidean center, but is offset toward the edge of the half-plane.

Now that we know how to find linear distances and areas of triangles, we can find the circumference and area of a circle using the same trick as Archimedes, approximating the circle by inscribed and circumscribed n-gons and taking limits. As noted on the preceding page, there is no concept of similarity in hyperbolic geometry, and so it is not surprising that the formulas for hyperbolic circumference and area aren't simple proportions, as in the Euclidean case. In hyperbolic geometry

• Circumference = 2 pi sinh r
• Area = 4 pi sinh2(r/2)
where r is the radius of the circle.

The applet

To help get you familiar with hyperbolic circles, 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 at the center of a circle and a blue point on the circle. The points are connected by a (hyperbolic) line segment, the radius, in red, and the (hyperbolic) circle itself is drawn in blue. Click your mouse on a point and drag it (while holding the mouse button down) to move the point. The radius and circle will follow the point. Off the edge of the half-plane (marked in gray), you will see the hyperbolic distance between the red point and the blue point. (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

• How does the circumference of a hyperbolic circle of a given radius compare to the circumference of a Euclidean circle with the same radius? Does it make a difference if you compare big circles or little circles?
• How does the area of a hyperbolic circle of a given radius compare to the area of a Euclidean circle with the same radius? Does it make a difference if you compare big circles or little circles?
• How quickly do the circumference and area grow as the radius grows? How does this compare to the Euclidean case?
• Note that the circumference always exceeds the area for a hyperbolic circle. Can you prove this using the formulas given above? Does this happen for Euclidean circles? Could you make it happen for Euclidean circles if you used mixed units (e.g. measuring circumference in millimeters and area in square kilometers)?