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I think a point concerning the mapping may have been missed. It
is a mapping from one space (Not a plane) into another space. A pinhole
because of its infinite depth of field maps all of the object space on to
the plane of its screen, but the screen can be in an infinite number of
positions. Thus any point in the object plane is mapped to an infinite
number of points in the image space and vice versa. (All assuming perfect
geometric optics.)