Wayne, if a DX camera sensor has the same number of pixels as an FX, wouldn't the picture be exactly the same size?
Tricky question.
My argument says No.
I am hoping to speak to all, not just to you, so please don't take offense when I state something you know.
On print paper, there is no debate, pixels are just a resolution factor of reproduction - Not an enlargement factor. Which is tricky to say, it's correct, but I feel my words are not the best possible choice, because of course printing 150 dpi is a 2x larger image than 300 dpi, but it is the same pixels, and the resolution is lower because it is more enlargement of the sensor size.
Which is actually true of the video screen too (more at bottom).
Bottom line, we are enlarging the lens image that might be DX 24x16 mm size, or it might be FX 36x24 mm size. Same as film in that respect - the sensor has size. This lens image is what it is all about. Pixels or pixel resolution or film grain do not create image detail, they just try to reproduce lens detail well.
For DOF calculation purposes, we are enlarging it to a standard 8x10 inch size on paper (we have to crop it from 8x12 inches for 3:2). The 8 inch dimension is left intact.
We are enlarging 16mm or 24 mm to be 8 inches on paper, which is roughly 12.7x or 8.5x enlargement without additional cropping, which enlarges any blur too. That is a 1.5x factor. We obviously see the enlarged DX lens blur larger and more clearly than from the same lens on FX.
Having more pixels will sample the lens image more accurately, and can increase printer resolution of printing dithered ink dots, but it does not affect enlargement of the original image size. The 8x10 inches determines enlargement of sensor size. It works exactly the same as at the darkroom enlarger. And FX and DX enlargements are like the 35mm and APS film sizes.
Circle of Confusion is the (arbitrary) hypothetical size of an out-of-focus point source dot in the image. We judge the maximum CoC size permitted to still call it "sharp" (if not visible). The CoC link above shows what this circle diameter is, in its first image at the top right. There is no abrupt cutoff of course, it is not sharp on one side and fuzzy across the line, it just gradually varies all along, but we state a numerical limit that can be calculated.
We necessarily must adjust the CoC limit used for degrees of enlargement. To be the same visibility criteria, it has to be larger at low enlargement, and smaller at high enlargement. That depends on sensor size.
History has had several estimates about the CoC limit we can see. Carl Zeiss is said to have defined it as film diagonal/1730, which is of course about sensor size and enlargement. Now CoC is commonly said to be 0.03 mm diameter for 35mm (FX), and 0.02 mm for DX (2/3 size, which is 1/1.5)... when enlarged to this standard 8x10 viewed at 10 inches. And only 0.004 or 0.005 mm is acceptable CoC for compact camera sizes, which have to be enlarged so much.
An
online DOF calculator will show the CoC that they use for the sensor size you select. Sensor size is the first question they ask us regarding DOF.
On the video screen, which shows pixels (but only up to about 2 megapixels), I say it is obviously the same thing. We do use pixels to do it, but we are still enlarging the sensor size to show at some viewed size in inches. About like paper, different plan, but same result of visibility. We resample to use as many pixels are necessary to create that size. It is an enlargement of the original lens image size, and the pixels are a resolution factor of reproduction of that enlargement.
Hope that helps.