For reference, here are shots containing a couple of initial designs. I'm not set on the full circle on the smaller one. I don't know that it's really necessary there, but it may help with holding and moving it around. The larger one is a 6" base and the smaller one is 3". They'll fit a Mitutoyo dial gauge, if all goes well, and allow for about 3/8" of depth measurement. Should be plenty on a small mirror.
Calculate the deepest sagitta (depth or height of a mirror surface) you're likely to encounter with your spherometer and then make the legs adequate for that. You'll be able to measure smaller depths/heights easily with that.
For instance, say that for now you'll most likely see a mirror that's a 10-inch clear diameter D, with a radius of curvature R of 60 inches (f/3). Your sagitta even with a spherometer spanning 8 inches will be less than 3/16". Most people, especially less experienced ones, are unlikely to make a 10-inch f/3 mirror. If you're 3D printing your spherometers, it's just as easy to make one with larger ball contacts, in case you decide to make something with deeper surfaces, like a Maksutov corrector. :o)
One thing to keep in mind is to make the spherometer body as robust as and stiff as possible to avoid flexure, and from the looks of your illustrations, you're already doing that, so !
For parabolic mirrors used in Newtonians a precise radius of curvature is not critical, however for Cassegrains and compound configurations, the radii of curvature have to match meet tight tolerances. Also, if possible, calibrate your spherometer against a certified surface.of a known radius of curvature
Every spherometer has two internal errors. One error is the in the distance between the ball contact. This distance is measured mechanically (usually with calipers) with an uncertainty of anywhere from 0.005 o 0.01 mm, depending on the model used. 3D printers operate at an accuracy of 0.1 mm, possibly more. The second source of error is your depth gauge/dial indicator.
In order to calculate your internal (repeating) spherometer error you need to take the error of the spherometer radius (dr) and that of your dial indicator into account (ds). The latter is usually found in the indicator's manufacturer's literature and is typically of the order of ±02 to 0.002 mm); again more expensive gauges may do better than that..
The error fraction contribution to the radius of curvature by the radius of the spherometer contact points Δr = (r/s)*dr, and of the dial indicator spindle, Δs = ((s/r)² − 1)*ds/2. The total reading error contribution of your spherometer then equals Δr + Δs. *(see reference below)
This then is your uncertainty factor in calculating the radius of curvature of a test mirror/lens measured by the spherometer.
Let's say that your spherometer r = 50 mm, dr = 0.1 mm, sagitta depth s = 1.25 mm, and sagitta error contribution ds = 0.002 mm.
Then Δr = (r/s)dr = 4, and your Δs = 0.5((s/r)² − 1)ds = 1.599, hence the total spherometer error contribution is 4 + 1.599 = ± 5.599 ≈ ± 5.6 mm.
Based on the sagitta of 1.250 mm and spherometer radius of 50, your radius of curvature R = (r² + s²)/2s = 1000.625±5.6 mm. The true R is somewhere between 995.025 and 1005.6 mm.
The actual R is 1,000 mm, but the reason you are getting 1000.625 is because your measurements have inherent errors in them even if your measuring techniques are flawless. In order for your R to read 1,000 your sagitta should have been known down to six decimal places -- 1.250782 mm! Still your results are very much in the acceptable tolerance envelope: (1000/1000.625)*100 = 99.94% accuracy.
I don't know what kind of thermal expansion and contraction is expected from 3D printed objects. That should have to be taken into account by recalibrating the spherometer before each use. The same is true of precision machined metal spherometers.
*Ref: William M. Browne, Advanced Telescope Making Techniques, Mechanical, Willmann-Bell, 1986, p.109
Edited by MKV, 05 November 2021 - 11:00 AM.