Fisheye images are a way of representing a photograph with an angle of view greater than 180°. Stereographic is a conformal cartographic representation that has similar properties but with less dramatic distortion: http://en.wikipedia.org/wiki/Stereographic_projection
This experimental tool (requires the Imager and Parse::RecDescent perl modules) converts from fisheye format to stereographic:
The following pair of images show a partial 288° fisheye view created in hugin and then converted to stereographic.
The following image is a full frame fisheye with 360° horizontal angle of view.
The process also works for partial fisheye views. The following picture covers only 160° horizontal field of view, even so it shows heavy distortion at the sides which are corrected when converting to stereographic.
By using a 720 degree angle of view fisheye image as below (simply a 360 degree fisheye extended in the gimp by enlarging the canvas by 200%), the stereographic output has an angle of view or around 360 degrees.