Feathers of a duck are 3D objects projected onto a 2D space in a camera. How well they can be resolved relates to how many data points are captured in two dimensions. It's easier to use just one dimension when computing focal lengths and print sizes. But it's not more valid to do so. In fact, it would be less valid under certain circumstances. This would be unavoidably apparent if any cameras had a pixel shape or CFA arrangement which resulted in more detail along one dimension than along the other.Thanks for putting this into general context. To sum up, if your interest is knowing the number of data points or colloquially how many pixels you have on a duck, then what matters is the number of pixels in your sensor and also the focal length of your lens squared. But, if you are interested in the information content of those data points, or in practical terms whether you can resolve the individual feathers of the duck, then that scales with the square root of the number of pixels or the focal length of the lens.
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