Up: Isometries Between Models
The Poincaré Disk and the Klein-Beltrami Model
The Poincaré disc can be mapped to the Klein model by pulling the Poincaré disc directly up onto a unit sphere (centered
at (0,0,1), and then stereographically project the resulting hemisphere onto
the plane z=0 (through the top of the sphere). Finally, shrink by a factor of one
half
Step-by-step, the following movie shows us:
- An up-close view of the Klein model, with two chords as
geodesics.
- We back up to get a better view.
- A sphere appears, sitting directly atop the disc, along with some vertical projection lines.
The points in the disc will follow these lines, changing only their
z-component, until they are wrapped about the lower half of the
sphere.
- Once we have a hemisphere, the projection lines are changed to
allow for stereographic projection, from the top of the sphere (0,0,2) down
onto the plane z=0.
- We now have a disc of radius 2. By shrinking the disc by one
half, we now arrive at the Poincaré model.
- We now approach the disc to get a better view. Note that the
geodesics have changed.
Mpeg of the Klein model morphing to the Poincaré disc
Equations
Up: Isometries Between Models
Created: Jul 15 1996 ---
Last modified: Wed Aug 7 12:04:28 1996