When we view a photograph made by a pin-hole camera, we are viewing an image comprised of countless circles. These are projected on film by the pin-hole. Their size is a function of the diameter of the pin-hole. The circles are called “circles of confusion” because they juxtapose each other; thus their boundaries are indistinct.
It is the size of these circles that determine if the image will be perceived as “sharp”. If the observer sees disks, the observer will perceive the image as being fuzzy. If the circles are too small to be seen as disks, the observer will perceive the image is being in good focus.
The size of the pin-hole is the key. If too small, twin demons of interference and diffraction induce a fuzzy image. Also, if too small, the exposure time becomes too long. We enlarge the pin-hole to gain image brightness, and this enlarges the circles of confusion. Now we must abandon the pin-hole and substitute a lens.
What size circles of confusion? A disk viewed from 3000 diameters distance appears as a point. Thus a 1 inch diameter coin viewed from 3000 inches is perceived as a point without dimension. That’s 250 feet. That’s too stringent for photography because of the contrast of our media and viewing conditions. So we define the circle size as 3.4 minutes of arc, which works out to 1/100 of an inch in diameter viewed from 10 inches, or 2/100 of an inch viewed from 20 inches (reading distance). Converted to metric, it’s 0.5mm in diameter viewed from 500mm.