Up: Isometries Between Models
The Klein-Beltrami Model and the Upper Half Plane
Here we show a mapping from the Klein model to the upper half plane.
It is composed of parts of two other mpegs. We go from the Klein disc
to the hemisphere (as in the Klein/Poincaré movie, and from there proceed to the Upper Half Plane, as
in the Poincaré/UHP mpeg. Scene-by-scene:
- Start looking straight down on the Klein model. Geodesics
are chords.
- Back up to get a better view.
- A sphere appears, sitting directly atop the disc, along with
vertical projection lines.
- Points in the disc follow these lines up onto the lower surface
of the sphere.
- We now have the hemisphere model.
- Rotate it by 90 degrees.
- And now stretch the hemisphere to cover the half plane, using a
stereographic projection through the point (0,0,2) at the top of the
hemisphere onto the plane z=0, on which the (completed) sphere sits.
Mpeg
Equations
Up: Isometries Between Models
Created: Jul 15 1996 ---
Last modified: Wed Aug 7 12:23:59 1996