Fisheye Projection

In a fisheye projection the distance from the centre of the image to a point is proportional to the equivalent spatial angle. Details see below.

Commonly there are two types of fisheye distinguished: Circular fisheyes and fullframe fisheyes. However, both follow the same projection geometrics. The only difference is that for a circular fisheye the image circle fits (more or less) completely in the image whereas for the full frame type the image lies entirely inside the image circle. A circular fisheye can be turned into a full frame one if you use it with a smaller sensor/film size and vice versa. If you use a fisheye adaptor on a zoom lens you can do this by zooming.

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