The Poincaré Disk and the Upper Half Plane

The Poincaré Disc and the Upper Half Plane can be identified in several ways.

One involves circle inversion. In the mpeg below, we show the Poincaré disc being flipped about the boundary of another disc - the one centered at (0, -1) with radius sqrt(2).

Mpeg movie

Equations

Another involves stereographically mapping the Poincaré disc directly onto half of a unit sphere, turning the sphere on its side, and then stereographically projecting onto the plane z=0.

There's a lot going on in the following mpeg, so let's go through it scene by scene:

We start looking down on the Poincare model, along with two intersecting geodesics.
We then move back away from the disc to get a better view.
We stretch the disc by a factor of two.
The sphere and lines that appear are intended to help visualize the next stage: Points in the disc move along the straight projection lines until the disc is wrapped around half of the drawn sphere.
We now have a hemisphere. This is another model of hyperbolic space, though not a standard one, and thus we do not cover it in detail. The geodesics are the intersections of the hemisphere with vertical planes. The hemisphere model will come up again and again in these animations, serving as a midway stage in several of the transformations. (There is another non-standard model not covered in these pages based on a parabolic-shaped region of the plane.)
The hemisphere is now rotated by 90 degrees.
And now we flatten the hemisphere down into a half plane. We do this via a stereographic projection from the top point of the sphere, down onto the plane that's tangent to the sphere at its bottom point (from the point (0,0,2) down onto the plane z=0).
The result is the Upper Half Plane model of hyperbolic 2-space. The geodesics appear as semi-ellipses because of the angle at which we are viewing the plane - in reality they are circular.

Mpeg movie

Up: Isometries Between Models

Created: Jul 15 1996 --- Last modified: Wed Aug 7 11:53:30 1996