I think the aspect of this you truly don’t grasp is that the final output magnification (how big the subject is on screen or in print and how close you are to it) is fundamental not only to equivalence but to depth of field as well.I'm not sure I understand this statement: "if you crop all else being equal you change DoF."
When I first read this it sounded like you were saying that If I take a 3000 x 2000 px. image in photoshop and crop out a 1000 x 500 section of that image that the depth of field magically changes. I can understand how the perception would change if you change the viewing distance or enlarge the image, but why would the depth of field change simply by cropping an image? Certainly that image is already set in stone, so to speak, and cropping alone could not change the depth of field.
After re-reading what you wrote, I now interpret your statement to mean that cropping a full frame image down to the same dimensions as an APS-C image will produce the exact same image and depth of field. (all other factors being equal). That, by the way, is exactly what I have said in this discussion. But, is it really correct to say that you are changing the depth of field? Isn't the depth of field for that cropped portion of the image exactly the same as it was before the cropping? How could the depth of field possibly be changed by simply throwing away pixels?
In your first scenario where you simply crop out the center 1000 x 500 pixels of your 3000 x 2000 image and that center section remains the same size then the dof stays the same, however that is not equivalence because your image is now smaller. If you enlarge that center section to the same area of your original image then yes, the dof reduces. You are applying a crop factor of 0.31 so your dof reduces by that factor. That is if your uncropped image was shot at f4 the cropped portion when reproduced at the same area as the entire shot has 4 x 0.31 applied, or equivalent dof of f1.24.
In a real world situation like that you are talking about the ‘focal length limited’ situation. Do this to prove this to yourself. Take a picture of a distant small bird at f8, your dof calculator says at f8 and the focus distance you should have 18” of depth of field. When the entire image is on your screen the bird looks totally in focus. Now zoom in to 100%, that is the same as cropping or using a crop sensor. Suddenly only half the bird is in focus, what changed? Only the magnification. But if we agree that is the same as cropping or using a crop camera that proves that to all intent and in practical understanding the sensor size (or the amount of a sensor area you use) impacts the dof.
Don't forget dof is what is in acceptable focus. In the entire frame the bird is entirely acceptably sharp, yet when viewed at a larger magnification suddenly the tail feathers and one foot are not acceptably sharp.
Because aperture, focal length and distance to subject don't include other variables required to determine dof; magnification and 'acceptable focus'. It's like asking the question 2x?=?, your wording doesn't give anybody enough information to determine the dof. Find a dof calculator that only requires aperture, focal length and distance to subject to work out the dof, they all require more information than that, principally sensor size. Sensor size determines the CoC or commonly accepted 'acceptably sharp', and once you know that CoC, or factor for acceptably sharp, you can determine the rest. But it is all predicated on same sized output in your comparison.I'm also interested in your other statement:
"Depth of field relies on only two factors after you determine what is ‘acceptable focus’. The size of the lens pupil. The size of the subject as you are looking at the image output. THAT IS IT."
I certainly don't disagree with your statement as a scientific fact. But from a practical matter, how does this differ from saying (as I did) that depth of field is determined by aperture, focal length and distance to subject? Hardly anyone knows the size of the lens pupil of their lenses. You probably do, but most people don't. So why is it not acceptable shorthand to use the aperture and the focal length? Two things that people do know.
Similarly, why is "distance to subject" not an acceptable shorthand for "size of the subject as you are looking at the image output." Here, I understand that we may be talking about different things when we refer to "subject." I am thinking about the typical situation where one is taking a picture of another person. Generally speaking, how close the photographer is to that person (coupled of course, with the focal length of the lens) will determine the size of the subject as you are looking at the image output.
In other words, if I am taking a picture of my wife with a 200mm lens on a full frame camera, I have a general idea of how much of the frame I want to fill with her face and how much I want of the background to show. I adjust my distance from subject to roughly correspond to the size of the subject that I will be looking at in the image output. At the same time, if I want to decrease the depth of field for the background, I may adjust the distance between her and the background. Again, I'd like to know why this is not acceptable shorthand for your more precise technical definition.
Distance to subject in isolation tells you nothing about dof.
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