Take two arbitrary glass blanks, where the diameter of
the smaller is more than the radius of the larger, but less
than the diameter of the larger. We call the diameter of
the larger "M" and the diameter of the smaller "T." Thus,
1/2M < T < M. Construct an ellipse whose major axis is
M + T. The major axis is shown in the diagram to the
right.
The foci of this ellipse are determined as follows: the tool T is placed, successively, such that one edge rests on either end of the axis. Where the other edge of T intersects the axis, this defines the foci. The tool is shown to the right at the two extreme positions of its "orbit," the ellipse.
This defines an ellipse whose semi-minor axis is the square root of MT. Such an orbit is easily cut out of a piece of thin plywood, with a jigsaw. Make the cut a few millimeters inside the ellipse's line, and then sand down the extra wood to the exact line.
Now, M is centered on one of the foci, just as the sun is at one focus of a planetary orbit. Next, T is "orbited" around, inside of the orbit, on top of M, thus grinding it down. M will become convex and T will become concave. In the diagram to the right, we show T at the two most extreme positions of its orbit around M. Successively fine grades of abrasives are used to grind down the surface to the desired depth. Polishing is done with the standard pitch lap method.
If we use the same orbit thus constructed with an alternative tool blank of 2T, this will also work, since the center of this larger blank occupies all the relevant positions that the edge of the regular tool blank does. The circle on the right shows the size of such a derivative blank, for this particular orbit. The arrangement with this blank is called the "Derivative perexternal." Its tool is termed T' ("T prime").
This method will only work if the relative sizes of M, T and the orbit are determined by the arrangements set forth. If a wrong-sized M, T or orbit are used, the result will be a zoned surface, with no optical use.
Since T will tend to hang down off of M, when T is furthest from the center of M, this becomes a bit bulky, and, unless T is kept level, a circular and very pronounced zone will be ground around M. However, this will not effect the optical quality of T, which is what becomes the concave surface, in this case. At the end of the job, throw M out, and use T as the optical surface. To avoid this problem, it is recommended that the derivative case be used, since this problem does not come up with it. Another easy way to get around this is to use what we term a "negative" of the case, as we define below:
The "negative" uses the exact same orbit defined above. A
larger blank of diameter 2M is substituted for M, which
larger blank has a hole in its center of diameter M. (This is
because the distance from the first focus to the end of the
major axis which is furthest from the center of M is exactly
the diameter of M.) Now, the hole is centered at one focus
of the ellipse, in the same way in which M was, in the
earlier case, and the same T is orbited around inside the
ellipse. This is exactly the arrangement shown above, but
with one difference: The M is now empty space, while the
part of T's orbit which was empty space is now filled up.
Thus, we call this a "negative." Either T or T' can be used
to grind in the negative. Where T is used, we call it a
"Regular Perexternal Negative." When T' is used, we call it
a "Derivative Perexternal Negative." In this case, the larger
mirror blank becomes a concave surface, while the tool
become convex.
One interesting characteristic of the perexternal is that when its tool blank T is closest to the center of M, the amount of its diameter hanging off of the edge of M is equal to the amount of its diameter hanging onto M at the position where it is furthest from the center of M. This feature aids greatly in setting up a job, making it easy to test how accurately we have arranged our set up. Also, the fact that the edge of T must exactly lie upon the center of M, at the position of T's orbit where it is closest to M, aids the construction of any set-up. If these features do not "line up" in an arrangement, we must check to see that the orbit, the M and the T are the right sizes, since, otherwise, the set up will not work.
I have found that this will work provided the relative sizes fall within a certain margin of accuracy. I generally settle for about 98 or 99 per cent accuracy. This level of accuracy is easy to obtain for any amateur telescope maker, not to mention a machinist.