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Background: elementary optics of concave mirrors
The Hockney/Falco theory relies heavily on the purported use of concave mirrors during painting and thus we must take a moment to understand the basic optics of these simple devices.
Here's a side view of a concave mirror. Parallel light rays from a distant object at the left are reflected and meet at the focal point F. The distance from the mirror to the focal point is the focal length f.
Simple concave mirrors can be considered sections of a sphere, having center C. From elementary optics we know that the radius of that sphere, r, is twice the focal length of the mirror, that is, r = 2f. Be sure to note that the radius of the
sphere r is not the same thing as the radius of the mirror disk
itself! For instance, a small shaving or makeup mirror might be a mere 2 inches across, while the radius of the sphere from which it might have been cut could be three feet or more. The problems in the next section relate to the (unduly large) focal lengths f and thus radii of the spheres in the Hockney/Falco theory, not to the size of the mirror disk itself.
Summary: The radius of a sphere from which a convex mirror is cut is twice the focal length of that mirror; the diameter of such a sphere is four times the mirror's focal length.
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