[ATM] Figuring Volume

[Anthony Stillman <atmer@flash.net>]

Vladimir and Donald have been bouncing this back and forth for a while.  I
thought I would confirm their results and hopefully address the underlying
question, when is it worth it to fine grind parabolize before polishing
out?

The minimum amount of glass that can be removed from a spheroid to turn it
into a paraboloid requires that the center ROC be shortened and the edge
ROC be lengthened.  That is, dig a hole in the center and flatten the edge.
Without going into the details of integrals and derivatives and minima and
maxima, I present below a table of such volumes for a variety of diameters
and focal ratios.

For comparison the last column of the table is the volume contained in 2
microns of piston for that given diameter.  What one might expect to remove
to polish out.

How much volume is really removed?  I think someone who knew what they were
doing and or did it with care, would need only go a few percent over these
values to obtain a good figure.




Minimum volume (cubic mm) that must be removed to fully correct a spheroid.


 Diameter     f/3     f/4    f/5      2 um piston
  250 mm       37      16      8          98
  500 mm      297     125     64         393
  750 mm     1004     423    216         884
 1000 mm     2380    1002    512        1571

 419.1 mm       175      74     38         276
 (16.5 inch   0.0107   0.0045   0.0023    0.0168)




Anthony


breve tempus longum est imparatis
A short time is long for the unprepared.


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