The Reference Spheres
To restate the foregoing, the same paraboloid can be produced from three different spheres of appropriate radius, as shown in
Fig. 36. The three reference spheres used in testing
a paraboloid.
Fig. 36. The largest sphere, A, is converted into the paraboloid by deepening it at the center. The smallest sphere, B, is altered by wearing away glass around the edge. The center of this sphere, b, is also the center of curvature of the central zone of the paraboloid, the focal length of which is equal to Pb/2, or PF. By wearing away glass at both edge and central zones, the third sphere, C, is changed to the paraboloid, its 70-per-cent zone being all that remains of that sphere.
It is thus seen that, by whatever means a paraboloid is produced, there are three spheres of reference with which the optician can test the figure of the mirror's surface. There are an infinite number of other spheres, of course, but the three described are all
that need be reckoned with. The distance between the center of curvature (b) of the mirror's center zone and the point of intersection with the axis of the normal to any other zone is equal to r2/2R, where r is the zonal radius, and R is the radius of curvature of the
central zone. For a 6-inch f/8 mirror, the value for the edge and central zones is 0.047" (a and b, Fig. 36), the same as the mirror's sagitta. The center of curvature, c, of the 70-per-cent zone is located exactly midway between points a and b.
Now, by locating the center of curvature of each zone, and measuring the distances between them, we can determine by comparison with the values of r2/2R how closely the figure of the mirror approaches a paraboloid. In the application of this formula, however, both pinhole and knife-edge would have to be moved together. In practice, it is customary to have the pinhole remain stationary and to move the knife-edge only. In this case, the distances between the points of intersection with the axis (of the rays reflected from the several zones) will have been doubled. The formula then to be applied in determining the length of knife-edge travel is r2/R.
Although the points of intersection thus obtained (a, b, and c, Fig. 39) no longer mark the centers of curvature of the respective zones, this convenient expression will continue to be used in their connection. Theoretically, the stationary pinhole should be placed exactly at or alongside the center of curvature of the central zone, but no error is introduced as long as it is somewhere in the vicinity (see Chapter V, How to Use the Foucault Device). An advantage of thus doubling the knife-edge travel is that the error percentage is halved. The computed value of the r2/R formula (distance ab, Fig. 39) is known as the correction of the paraboloidal mirror, which for the 6-inch f/8 is 0.094". Ellipsoidal mirrors are referred to as under corrected; hyperboloids are over corrected. The paraboloid is a fully corrected mirror, and is the thin division between the two.
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