Re: [ATM] parabolic versus spherical volume of glass

["vladimir sacek" <v959@msn.com>]

Assuming that "differential volume" stands for the minimum volume
of glass needed to be removed, we do have some discrepancies.
I did my figures more precisely (I've being rounding a bit too much
previously), and came up with these volumes:
#1  0.0091 ci
#2  0.0027
#3  0.0065
#4  0.0093

One important correction is for parabola #3, which needs to have
taken about 2/3 as much from the edge as #2 (not 1/3, as I 
posted previously), and proportion of digging in the center vs. edge
is about 3 to 1 (not 6:1). Consequently, the depth of glass taken 
from the edge is ~D/6000F^3. Also, needed total volume of glass
to remove is more than doubled vs. #2.

The volume formula I use is V=Vp-Vs+x*Pi*d^2, where Vp is volume
of the parabola, Vs volume of the sphere, "x" edge-to-edge separation
between the sphere and parabola when they are touching at the 
given zonal hight, and "d" the aperture semi-diameter. The sphere and 
parabola volume formulas are identical to those you use, I believe, 
and x=(1/2Rs-1/2Rp)(1-z^2)D^2 + [(1-z^4)d^4)]/8Rs^3
with Rs and Rp being the respective curvature radii of sphere and 
parabolas, and "z" given zonal hight for which both sphere and
parabola have a common focus.

The bottom line I end up with is that the least amount of glass
removed is for parabola #2, approximately D^3/13,000F^3. 
#3 requires more than twice, and #1 and #4 more than three 
times as much glass removed.

Vlad
  FYI, for the example given in the spreadsheet (D=16.5, F/5), the
  values of fl diff for the parabolas described by Vlad are:
  parabola   fl diff   vol diff  min sag  Vlad's sag values for (D=16.5, F/5)
    #1         0       .00787       0%R   D/1024F^3=.000129
    #2        -.0515   .00198      70%R   D/4000F^3=.000033
    #3        -.074    .00351      85%R   D/13,000F^3=.000010
    #4        -.104    .00943     100%R   D/1000F^3=.000132

  Using the spreadsheet, #1, #2, and #3 agree with Vlad's numbers
  fairly well.  For #4, the point at which deepening the center is
  D/1000F^3=.000132 is at fl diff=-.104, but the glass removed is
  20% more than #1.  The point at which glass removed is 20% less
  than #1 is at fl diff=-.091 where the deepening of the center is
  .000099
  Vlad?

  Note that the plot is for 20 intervals, so the resolution of zones is 5%R.
  The spreadsheet is at:
  http://www.atmlist.net/contrib/donald-dot-good-at-comcast-dot-net/index.htm<http://www.atmlist.net/contrib/donald-dot-good-at-comcast-dot-net/index.htm>

  Don

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