I was fooling around again recently with some "Wee Planet" stereographic panoramas (previously mentioned here), and came across some tutorials on doing this with Photoshop
/GIMP using a Polar Coordinate distortion filter.While this will give you a similar result to what you might see from a true stereographic projection, it is not quite as aesthetically pleasing I feel. This is because the Polar Coordinate transform is not a true stereographic mapping, and will tend to squish objects in a very non-pleasing way.
I've assembled a few examples to demonstrate the effect and for comparison. The examples here are from Flickr user gadl (Alexandre Duret-Lutz) and are Creative Commons BY-NC-SA. Thank you Alexandre for making these available!
And really, what better examples to use than from "La Ville-Lumière", Paris!
