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------------------------ Optics Primer - Fisheye Projection Lenses

--------------------------------------------------------- By Bill Claff

If you own a lens with a fisheye projection there is a good chance you don't know the actual projection.
That's OK since you're unlikely to be using your lens for scientific measurements.
Probably you were more concerned as to whether it is a circular or diagonal fisheye; which is a matter of focal length and image plane size.

If we have an optical prescription (or certain data from the manufacturer) we ought to be able to place a lens in the spectrum of various fisheye projections.

I will use the Nikon Fisheye-Nikkor 6mm f/2.8 as an example because it is the only lens for which I have both a manufacturer provided table and a (likely) optical prescription.
The following chart tells the whole story:

I chose the y-axis spacing to make the Equidistant grid obvious. This lens is supposed to be Equidistant.
The red dots are from the data provided in the Nikon User Manual.
The blue open circles are from the Optical Bench using the optical prescription.

Agreement between the Optical Bench and the Nikon User Manual is good; both indicate an Equidistant projection.
The closest other fisheye projection, Equisolid, is shown for comparison.

The Nikon data makes it look as though there is slight barrel distortion toward the edges.
The Optical Bench data actually looks slightly better starting off with mild pincushion distortion and finishing with mild barrel distortion.

We should not be surprised at the results but a confirming datapoint is reassuring.

Here is test data from the Optical Bench for 27 fisheye lenses in the context of the Projection Formulas:

And, for clarity, a closer look:

I added a green dashed Angle Boundary line to indicate how angle is constrained by sin ( Θ / k)

The majority of these lenses are clustered fairly tightly together and are Equisolid projections with approximately a 180 degree angle of view.
It would seem that this is the type of fisheye projection that most photographers desire.

Five lenses appear to be Equidistant. The lens at 180 degrees directly on the Equidistant line is a Russian lens that I believe was specifically designed for aerial photography.

The stereographic and orthographic lenses on this chart happen to be from the automotive industry.
I have been unable to locate examples for Interchangeable Lens Cameras (ILCs).
Note that one stereographic lens is not so wide angle but is clearly stereographic as opposed to rectilinear.

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