Originally posted by photoptimist This is exactly true. The optical refraction of light, the convergence of the rays behind the lens, and any blur if the film or sensor aren't at the exact point of focus is only a function of the focal length and the various distances from subjects to lens to focal plane. The width and height of the film or sensor have no influence on the size of the circle of confusion. The lens does not know how big the film or sensor is.
But that has nothing to do with equivalence (except to create a lot of arguments about using the same focal length lens on different formats zooming with the feet etc.).
The effects of equivalence arise when one wants to take truly equivalent images in two different formats.
Imagine I see some beautiful 16x20 image taken on a 6x7 with a wide-open 105/2.4 and I want to make the equivalent 16x20 image (with identical foreground-subject-background perspective, identical framing, and identical blurring of out-of-focus regions) but using a K-3.
1) To get the identical foreground-subject-background perspective, I need to put the K-3 in the same location as the 6x7 was in with all the various lens-to-subject distances being the same.
2) To get the identical framing, I can't use a 105 on the K-3 because it's too narrow. I need to use a 30 mm lens on the K-3 (30 = 105 * 16mm APSC_frame_height / 56mm 6x7_frame_height ) to get the equivalent angle of view for getting the equivalent framing.
3) To get the identical blurring of out-of-focus regions with a 30 mm lens on APS-C as a 105/2.4 produces on 6x7, a messy bit of math reveals that I need a 30 f/0.7. (The reason for this is due to the smaller optical magnification of a 30 mm lens on APS-C and the effects of having a smaller circle of confusion on APS-C. That forces me to need a physical aperture on the 30 mm lens as big as the physical aperture on the 105/2.4. That's a 44 mm diameter physical aperture which then is equal to a f/0.7 relative aperture on a 30 mm lens. The overall result is that the ratio of the format sizes determines the ratio of relative apertures needed and that 105/2.4 on 6x7 is actually a very shallow DoF lens compared to a 30/2.4 on APS-C.)
Equivalence is about attempting to make equivalent images on different formats and has nothing to do with the same lens or focal length looks on different formats.
I think if you’ll read my later post, you’ll see that I added the effect of enlargement as an offsetting effect. And you’ll also note my care in distinguishing between “aperture” and “f/stop” or focal ratio.
But that does not mean the optical magnification is unimportant. Blur is controlled by absolute aperture and magnification. The magnification may be in the lens or in the “enlarger”. When print size is controlled, enlargement magnification required by a smaller format undoes some of the increased depth of field resulting from less lens magnification (resulting from a shorter focal length used with a smaller format).
Of course, wanting large prints is one reason people use larger formats, so the assumption of equal print size is also a model rather than a reality.
But, as you say, the first step to mathematical understanding is the distinction between aperture and focal ratio. That’s why large format often uses very long shutter speeds—the absolute aperture needed for acceptable depth of field with the longer lenses results in a much higher focal ratio—f/22 instead of f/8, etc. But the focal ratio is what controls the intensity of light and therefore the exposure.
The reason I know this stuff? 1.) I work in multiple formats from APS-C (and smaller) to 4x5, often adapting lenses and cameras to work across formats. 2.) As an engineer, I’m not afraid of math. 3.) As a teacher, I relish opportunities to explain things, but only things I actually know, which requires a willingness to learn stuff that works from first principles. 4.) And I’ve been involved in various photographic forums since the late 90’s, where these topics come up routinely.
I learned about lens calculations from optical texts, mostly. Kingslake, etc.
And one learns a lot enlarging various formats using various lens focal lengths and a good grain focuser.
Rick “who has done the math—years ago” Denney