The Paraboloid
Fig. 35. Three methods whereby a sphere may be altered into
a paraboloid. In each case, the graph at the left shows the
departure, in millionths of an inch, of the paraboloidal surface from the same 6-inch f/8 mirror.
........surface; the new focal length is equal to one half the radius of curvature of the deepened central zone. Of the original sphere, only an extremely narrow edge zone remains as part of the paraboloid. In mirror work, this is the usual method of parabolizing, done by polishing with the mirror face down on the full-sized lap.
In b, the bulk of glass is removed from the edge zones of the sphere, tapering down to zero approaching the center, where the original sphere merges with the central zones of the paraboloid. In this case there is no change in focal length, as the central zone may be regarded to have been untouched in the operation. An equal amount of glass is removed by this method, but because the work has to be performed with the lap on top, inevitably leading to turned-down edge, it is seldom attempted.
In c, glass is removed from both edge and central zones of the sphere, its 70-per-cent zone (a zone the radius of which is 70 percent of that of the mirror) remaining as part of that paraboloid. The least amount of glass is removed by this method, which also has to be performed upside down, but with a small polisher, used locally. In this operation, the focal length is lessened by half the amount of reduction made in a.
Of course, the mirror sections represented in Fig. 35 are enormously large by comparison with our 6-inch f/8, which occupies so small a portion of the paraboloidal surface (see Fig. 37) that it and the sphere are indistinguishable from each other by any ordinary means of measurement. In the graphs at the left of the diagrams, however, the departure from the sphere, measured in millionths of an inch, is scaled to the apparent magnification of about 100,000 obtained with the Foucault test. Note that the differences at a and b are identical — amounting in the case of a 6-inch f/8 mirror to 11.4 millionths of an inch.
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