Because that's what the word compare means. Duuuuh. An analogy: Let's compare the suitability of two different woods; poplar and cherry. Let's use the cherry to make a musical instrument -- a wooden flute, and the poplar to make a kitchen cutting board. And now we compare the suitability of the woods. Can you see where we went wrong?
You still don't get it do you? You're saying
y=x but in your proof you're specifying that the condition y=x must be met to be a valid comparison. I'm not making kitchen boards out of sugared cherries but explaining the flaws in your theory because you seem hell bent on linking smaller sensors to greater dof when all the evidence says it is the smaller effective aperture diameters that create greater dof, the smaller sensors do
not always have greater dof.
I've shown you where your model breaks down, I've shown you where larger formats have greater dof, I've shown you the two variables that are consistent with increasing and decreasing dof and yet you still dismiss them as invalid because they do not fit your theory.
n other words show us a DOF calculator that doesn't account for sensor size in the MATH used to calculate DOF.
Of course dof calculators account for sensor size!!! c is directly defined by dividing the finished print size by the sensor size, that's how you calculate
c! But please note that the smaller the sensor then the less dof for any given COC, by your very own maths as you've presented???
Same hang-up, you're trying to calculate DOF at the sensor.
NO!!!! I am not trying to calculate dof at the sensor at all. I'm fully aware of dof and what it is.
Basics.
Light is reflected of two point objects at different distances to the lens. The lens then bends each "ray of light" by an equal amount. If you adjust the focus so the light from the distant object is focussed on a point on the sensor then the light from the nearer object will be focussed beyond it creating a "circle of unfocussed light at the plane of the sensor (COC).
These are point light sources focussed on a plane. Fov does not come into the equation here. The COC formed at the sensor is independent of sensor size.
The image at the sensor is then magnified to produce the final print at the designated size for determination of dof, etc...
Now I'm completely failing to see how a smaller sensor can possibly produce greater dof here. With my maths the smaller sensor needs more magnification, and the more you have to magnify an image to produce a 10"x8" print then the more you'll magnify the COC s formed at the sensor and reduce dof.
Yet within a range of focusing distances (as explained before it's really noticeable at portrait distances for landscapes the larger formats can have either more or less dof, in short they are more adjustable) smaller sensor cameras appear to have greater dof (appear because dof is an illusion).
So what causes it? If reducing sensor size has a negative effect on dof,
which it clearly does, then there must be something else at work.
One thing controls dof: the more parallel the light from the two point objects at different distances is when entering the lens then the more parallel it is when leaving the lens and thus the smaller the COC produced at the sensor is for the out of focussed object.
Two things affect this and neither are sensor size. The more distant the objects are then the more parallel the light coming from the two point object will be entering the lens (just move both objects back away from the lens in my above diagram and see what happens).
The smaller the effective aperture diameter is in front of the lens then the more parallel the rays passing through them lens from the two point objects will be (try introducing a smaller aperture in the diagram above and see what happens).
Sensor size does not affect any of this. In fact all I see is that the smaller the sensor is then the more you have to magnify the COC produced on the said sensor thus reducing apparent dof. To me this is simple and plain.
Now with your example of same photo and same exposure. You set the conditions of constant fov and constant
f-stop. You will not compare
f5.6 against
f5.7 or an fov of 36 degrees against 35 degrees. You specify and set the condition of the test that comparison must be made only when the effective aperture diameter is adjusted in exact proportion, and so to compensate, to the change in sensor size. Then you tell me that I cannot rule sensor size out of the equation because it's a condition that you specify must be met before you even evaluate the results.
(To achieve the same fov on different formats then you must reduce focal length. f-stop (exposure) is the ratio of the focal length to the aperture diameter. So to maintain fov across sensors you must reduce the effective aperture diameter in exact proportion to the sensor size. This is the condition that you have set, this is the condition that you will not deviate from, this is the only condition that you will accept as a valid test and because the condition of the test you have set inextricably linked sensor size to effective aperture diameter you say you've proved that smaller sensors have greater dof.)
So what happens when you don't meet that specific proportion? What happens when the photos on the two sensors are not exactly the same and are just slightly different, (much like real photography)? What happens when we change sensor size while maintaining the effective aperture diameter? (
Smaller sensors have less dof). What happens when we reduce the effective aperture diameter while maintaining sensor size? (
Dof increases).
Can you see a simple rule that holds true in all cases yet? (
Dof is proportional to the subject distance and effective aperture diameter, [distance, f-stop and focal length], not sensor size).
A resounding, emphatic TRUE! in the 1970's Kodak in point of fact, invented an entirely BRAND-NEW, ultra-tiny negative area film FORMAT, to get hyperfocal depth of field that extended from approximately 3 feet to Infinity, at around f/8, with the incredibly teeny-tiny Kodak Disc Format.
Yes, but as I stated before, once you get passed this hyper-focal distance then the dof of the smaller sensor does not improve, yet the dof on the larger sensor does improve. So at
f8 with the closest subject at a distance of say 300' (landscape) to infinity is the extra dof of a smaller sensor an advantage, or would you actually get better results from a larger sensor that needs less magnification to the finished print?
This is the very real limit of smaller sensors, that when you hit that hyper-focal distance there is less improvement in stopping down than you get from a large sensor. The gap closes and in some instances the larger sensors are better because they require less magnification. I ran into this wall very often when I briefly used APS-C cameras. Not because I wanted to increase dof but because with the kit lens once your subject exceeds the hyper-focal distance it is very difficult to decrease dof!