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