photo by Alexandre Duret-Lutz
A stereographic projection is technically mapping a sphere onto a plane. What photographers like Alexandre Duret-Lutz, Masato, Seb Perez-Duarte, Boltron and Masakazu Matsumoto have done is take a 360° panorama photograph that covers -90° (the ground) to 90° (the sky) and then mapped that onto a plane.
photo by Masakazu Matsumoto
Regardless of how it is done, the effects are beautiful.
photo by Masato
Masakazu Matsumoto provides a FAQ on the technique at his flickr set.
via Pink Tentacle