Fisheye Projection

in a fisheye projection the distance from the centre of the image to a point is proportional to the equivalent spatial angle.

Taken from a posting of Helmut Dersch:

The focal length f of common fisheye lenses corresponds
quite simple to the angle of view theta and the
radial position R of a point on the slide:

R = 2 * f * sin( theta/2 )

So for 90 degrees, which would be the maximum
theta of a 180 degree lens, f=8mm, you get
R = 11.3mm, which is the radius of
the image circle.

This projection model applies to the Nikon 8mm
and the Sigma 8mm (which actually has f=7.8mm).
This is also what you get when you look into
a convex mirror.

Some older Nikon lenses (e.g. the 7.5mm) try to
approach a linear mapping

R = f * theta      (theta in rad).

and succeed more or less.
For most practical applictions, you won't see a big
difference between the two.

Btw, a rectilinear lens has a mapping

R = f * tan( theta )

We can assume that most newer fisheyes follow the first mapping scheme.

Complete text of the mail can be found at W.J. Markerink's page about fisheye analysis

More information on fisheyes and their distortions in this PDF from coastal optics