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Well to be pedantic, the format that gets the most out of a circle of acceptable quality is a circle, not a square.

Point taken. That makes me curious if any cameras were ever produced in that format. I would suspect that circular format photography might be used in astronomy where every last bit of the image is valuable.

Yo’re right though, a square is certainly the rectangle of largest area from a circular lens.

If there was a need to maximize the capture for technical reasons I imagine it would be easier to just oversize the film or sensor and trim the corners.

There's a number of fisheyes that produce circular images on rectangular film (obviously, not using the entirety of the film surface).

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