Building a spherometer

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To be able to measure the focal lenght of a mirror I am grinding I built a simple, inexpensive, but acurate spherometer. I used the instructions on Bob May's website to build it. It is, as Bob calls it, a ‘2 leg spherometer with an outtrigger’.

Parts for the upper spherometer

1. A dial indicator. I bought one for €25,95 (= about $32,-). Accuracy: 0.01 mm (0.0004").
2. A piece of 6 mm thickness aluminium bar, 20 mm wide and about 150 long (lenght depending on the diameter of the measuring circle one has in mind).
3. About 100 mm of 6 mm thickness metal tubing (I used stainless) to make the legs.
4. Three small metal balls fitting on the metal tubing, to act as the feet of the legs, resting on the mirror.
5. Two bolts to clamp the dial indicator in place.
6. Some metal epoxy to glue the legs into the holes in the aluminium bars and to glue the balls to the legs.

Large spherometer

The legs of the spherometer in the upper photograph are relatively close together. For a larger mirror it can be necessary to make one with the legs further apart for better resolution. I also made such a wider one, useable for 12 inch and larger diameter mirrors. Disadvantage of a large one (compared to the mirror diameter) is, that it is much less (or not) possible to take readings over the whole surface of the mirror. This time I used bolts and wingnuts for the legs and glued the balls to the bolts. The length of the legs can now be adjusted. For this large spherometer a 6 mm thick aluminum bar is not thick enough. The bar will bend and the shperometer will measure very inaccurately. Therefore I glued a piece of 20x20 mm aluminum tubing to the bar between the outer two legs. A photograph is below.

The spherometers work very well. With the small one, I made two pairs of mirrors for binoculars. Two 12 inch f5 mirrors, with focal length's only 1 millimeter difference and two 8 inch f/6 mirrors, differing 3 mm in focal length.

Costs and dimensions

Total cost of each spherometer: about €34,- ($41,-). The distance between the two legs on the same bar as the dial indicator is the diameter of the measuring circle of the spherometer (diameter of mine is 90.8 mm for the narrow and 240.8 for the wide legged spherometer). The third leg, also resting on this circle prevents the spherometer from falling and holds it in the corect measuring position.

Formula

The focal length of the mirror is calculated with formulae: F = 1/2(s^2+r^2)/2s, in which F is the focal length, s is the value as measured with the spherometer and r is the radius of the spherometer's measuring circle, or half of the distance bewteen the two legs on the same bar as the dial indicator. More exact is a formula where the diameter of the balls on the spherometer legs are taken into the account. Formula: s=(R-d/2)-((R-d/2)^2-r^2)^0.5 for concave surfaces and s=(R+d/2)-((R+d/2)^2-r^2)^0.5 for convex surfaces, in which s-sagitta, R=radius of curvature, d= diameter of balls and r=radius of spherometer. Nice and easy approximations are: s=d/(8*R) for sagitta and R=d/(8*s) for radius of curvature in which s=sagitta, d=diameter of spherometer and R=radius of curvature (from Ray Williamson (2009), Handy formulas).

If you make one: be sure the dial indicator cannot move in the aluminium bar, otherwise it will measure inconsistently.

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Email to: Jan van Gastel