A long time ago I wrote:

>>> However, the sizes of objects in the image *is* accurately predicted by

>>> the focal length ratio, all the way to the widest wide angle, as long as

>>> you're still talking about rectilinear lenses (not fisheyes).
"David Ruether" <(E-Mail Removed)> writes:

>> How can this be? If you shoot a very large grid of small squares from a

>> constant distance (assuming constant sensor size and distance - and with

>> the subject grid parallel with the sensor), and keep halving the lens focal

>> lengths, the angles of view will progressively not follow the FL reduction

>> ratio (as I showed earlier), resulting, I would think, in less than the

>> number of squares visible in any direction in the image than a straight

>> following of the ratio would predict. In other words, the size of subjects

>> as rendered in the image is also not proportional to the FL ratio when it

>> comes to short FL lenses (they are larger...). Am I missing something?

>Ah, I just finished a MUCH more detailed coverage of this, which

>is going up now on my web page, listed in the "Articles" page as,

>"On Lens Angles Of View, Magnification, And Perspective. The

>direct URL is --

>http://www.donferrario.com/ruether/l...erspective.htm
Sorry I didn't comment sooner - I never saw this reply. But you are

wrong when you say "resulting ... in less than the number of squares

visible ... than a straight following of ratio would predict".

Say you go from 36 to 18 mm lens focal length. The FOV goes from 62.86

to 101.32 degrees, less than a factor of 2 increase. But, as long as

you are shooting a flat wall with a camera aimed perpendicular to that

wall, you will still get twice as many squares visible. The true field

width, on the plane containing the grid, *is* doubled.

If you don't believe me, draw a scale diagram and measure. Or do the

math using similar triangles. You've decreased the lens-image distance

by a factor of two, while keeping the image width the same, so the

tangent of half the FOV has *exactly doubled*, even though the angle

itself has not doubled. On the subject side, the triangle is similar

to the image-side triangle, so the subject-side tangent of half the FOV

is *also* doubled. But the lens-subject distance has not changed, so

the amount of the subject visible has exactly doubled.

Or the handwaving argument: you haven't doubled the FOV angle, but the

additional squares you can see are being increasingly foreshortened by

the very wide angle they are off-axis, so you get more of them in each

degree of extra visual angle. The two effects cancel, and you get

exactly twice as many squares in not twice as many degrees.

By the way, your argument would be correct if the squares were drawn on

a sphere centered on the lens, since the squares would always appear a

certain number of degrees wide everywhere in the field. But we're

assuming a flat subject, not a spherical one.

As it happens, I write camera software for video games, so I can create

any wideangle lens I want, no matter how extreme, as long as it is a

distortion-free rectilinear lens. When I want, say, "50% more field of

view", I do *not* multiply the camera FOV by 1.5. I divide the

effective lens focal length by 1.5 instead. (This is actually

implemented by multiplying the tangent of half of the FOV angle by

1.5). This gives the desired effect: making all objects 2/3 the size

they were, and showing 1.5 times more width of any plane perpendicular

to the line of sight - even though it does not double the FOV angle.

I just recognize that FOV is nonlinearly related to what I really want

to control, which is magnification.

Dave