stereographic projection in fisheye lenses

introduction

stereographic projection is a way of picturing a 3D sphere as a 2D plane to maintain angles, and this doesn't compress marginal objects as much.
stereographic projection has been used to map 360 degree spherical panoramas and Little Planet imagery
- this results in effects known as a little planet (when the center of projection is the nadir) and a tube (when the center of projection is the zenith).
traditional fisheye lenses use an equal-area projection in which areas close to the edge retain their shape, and straight lines are less curved, these images can be automatically converted to stereographic projection using tools such as Panotools
some newer fisheye lenses such as the Samyang / Rokinon 7.5mm f/3.5 manual focus fisheye for Micro Four Thirds use stereographic projection which is more expensive to make