Uploaded November 2013 | Updated September 2026, 1 week ago

---

Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each).

spherical projections:

1. orthographic projection of the sphere (at 17s),

2. gnomonic projection of the sphere (at 1m16s),

3. stereographic projection of the sphere (at 2m33s),

4. Lambert azimuthal equal-area projection of the sphere (at 3m59s),

5. azimuthal equidistant projection of the sphere (at 5m23s);

hyperbolic projections:

6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic,

7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic,

8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic,

9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s),

10. azimuthal equidistant projection of the hyperbolic plane (at 11m55s).

The spherical projections show the sphere with a dodecahedral tiling by regular pentagons; the hyperbolic projections show the hyperbolic plane with a uniform tiling by regular heptagons.

The program used to compute the frames is available at madore.org/~david/programs/#prog_projections

For the same projections of the Earth's surface, see: youtube.com/watch?v=LKcTbIsWS9c