I think this is how vignetting happens

The diagram above indicates the dimensions of a 5" x 4" paper negative as used in my pinhole cameras. I have been trying to fully understand why there is darkening at the corners and how to reduce that to a minimum without having to loose too much of the great wide-angle ability of pin hole cameras.
The triangle on the left show the dimensions as if on the centre line across the negative whilst that on the right indicate the dimensions as if on the diagonal across the negative.

You can see the circle in the centre indicating that only the very corners may show any vignetting .

The difference in lumination, I am using the cos power3 here not cos power4  left ..70%   that on the right 59%  so you can see there is a drop off just at that outside the circle part ... which also reflects the  difference in f number 284 (125/ 0.44) on centre point 318 (140/ 0.44) on outside edge of the centre line and 366 (148/0.44) on the outside edge  of the diagonal when using a pin hole of 0.44mm.

F284 x 1.414 = 401  so there is not even one stop difference between centre and outside ....in fact I guess that there is not even half a stop ( but I do not know how to calculate that, all I know is that a full stop increase by 1.414 the square root of 2  !!!


As some consider falloff to become perceptible when the corner illumination falls to more than 1 stop less than the centre, that is about 50% of the centre light level, so  I feel sure I am within that limit.

I hope you find that interesting ... and I hope I have it correct ...  well I shall also prove it empirically with photos !!