How to describe the type of projection of a lens?
It is very easy to do so by using a Optical Lens Design software when the detailed optical design and the construction of the concerned lens is publicly available. For example that is the case when a patent for the lens has been issued. But when detailed information is not made available about the different lens components and their arrangement, the only way that I could think of is a kind of reverse ingineering. Let's describe it.
1) Determination of the No-Parallax Point: Many ways can be devised (google search for it). I personally use an optical bench based on a LASER pointer.
2) Shooting step by step a series of photographs that shall allow to measure on each image the distance (in pixels) of a fixed point (in the object space) from the center of the sensor while the lens is revolved around the NPP with known discrete successive steps. .
3) Use a spreadsheet to collect all the data, make the required coordinate and unit transformations and in final draw the graph describing the projection. I have named "Image Radial Mapping of a lens" this whole operation .
A composite rotator system allows to rotate around the NPP (and around the vertical axis) by tiny angular steps. It is composed of a Manfrotto MA300 in conjuction with a D-16 rotator and an Ultimate R-1 head + D-4 rotator both from Nodal Ninja branded by Fanotec. The lens is clamped in a R-1 ring in such a way to have a diagonal of the sensor fixed in the horizontal plane. The image shall then later be peered at to explore along the diagonal, that is to say the Maximum Angle of View (i.e. from one corner to the oposite corner). For your information, the Canon 8-15 mm lens delivers a rectangular image with about 180 degrees of angle of view ( diagonal) when the zoom is set at the "15 mm" focal length mark.
At the front end of a long T shaped aluminum bar, there is a thin vertical pin (far left on the picture).The bar is attachedand fixed on the interface between the bottom of the D-4 rotator and the top of the R-16 rotator.
In a first step and with the help of the camera viewfinder, the pin is roughly aligned with the lens axis (i.e. the center of the image sensor) and it is simultaneously aligned with a small target object at a far distance: In a second step the live view with x10 zoom of the EOS 5D2 is used to better refine the alignment. Then finally, a cross-check on a test-photograph is performed to get the mechanical alignment as perfect as possible.
In the rest of the experiment, the aluminum T bar shall revolve together with the camera because after initial adjustment, the D-4 rotator is hard locked. The elevation/inclination of the bar was adjusted by shimming: some (green) paper wedge is inserted at the root of the cantilever beam i.e. between the bar itself and the D-16 rotator body cylinder .
The conjunction of the Manfrotto rotator with the NN D-16 rotator reduces the smallest possible step from 3.75° down to 2° (e.g. 20° -on the right- minus 18° -on the left-).
About 90 to 95 photographs are shot to cover a whole 180° of angle (or more) and eventually repeating for every focal length of the whole 8-15 mm range.
I later measure in Photoshop the distance in pixels that separates the target object from the pin at each angular step and use Excel to compute the image radial mapping (which thus represents the fisheye projection).
On the picture one can see the aliminum T bar from the side (left of pict) and the same vewed from the front of it.