ATM Sagitta

[Anthony Stillman <atmer@flash.net>]

All,

Not too long ago Richard Schwartz posted a messages under the subject
heading "ATM Measuring rough grind depth..."  In it he proposed the
following as a means of determining focal length.

>Lay a straignt edge across the mirror.   Find a drill that is small
>enough to fit loosely under the straight edge.  Slide the drill as far
>as it will go toward the edge of the mirror (without lifting the
>straight edge, of course).   Calculate the focal length from the mirror
>diameter, the drill diameter, and the distance from the drill to the
>edge.  The exact formula is left as an exercise for the student.

Well it sounded like a good idea to me and I took up the chalange.  I
played with the problem at odd moments over the last week and I can now
offer a solution.

First let me define the problem more carefully and introduce some notation.
Specifically, I am replacing the drill bit with a narrow gapping guage
which measures the distance perpendicular to the straight edge, from the
straight edge to the surface of a sphere.  I suspect a drill bit is a good
approximation of this if one uses just the very end.  But I want to point
out that I'm not dealing with the curvature of the drill bit and its actual
contact point with a curved surface of the mirror.

Notation

  R   - The radius of curvature of the mirror under test
  r   - The radius of the mirror under test, that is half its diameter
  h   - Distance from the edge of the mirror to the gapping guage or drill bit
  g   - The thickness of the gapping tool or the diameter of the drill bit

Since ASCII art isn't my strong suit, I'll stick to (hopefully) useful
explaination.  Anyone who has looked at a basic text on ATMing knows that
the standard sagittal formula for a sphere, namely  s=R-sqrt(sq(R)-sq(r))
is derived from the hypotenuse of a right triangle being equal in length to
the square root of the sum of the square of the two other sides. This
formula can of course be solved for R, namely R=(sq(r)+sq(s))/2s  A little
drawing and some small amount of inspection yields a similar but slightly
different formula for the gap g.  Specifically,

   g=sqrt(sq(R)-sq(r-h)) - sqrt(sq(R)-sq(r))

Alas, this formula can not be solved for R. However, it is possible to
re-write this equation and expand it into two series.  Then, to what ever
accurcy is desired, truncate the series and solve for R.  What I did was
pull an R squared out of the first and second root terms to obtain

   g=R*( sqrt(1-sq(r-h)/sq(R)) - sqrt(1-sq(r)/sq(R) )

Here the first and second terms are in the form sqrt(1-x) where for the
first square root, x is sq(r-h)/sq(R) and for the second square root, x is
sq(r)/sq(R)

The square root of 1-x expands to

   1 - x/2 - x^2/(2*4) - x^3/(2*4*6) - x^4/(2*4*6*8) ...

so long as x <= 1 which should be true in both cases for all mirrors except
f/fishbowl.

So to first order g now reduces to

  g=(2rh-h^2)/2R

This can then be solved for R

  R=(2rh-h^2)/2g

For h=r this reduces to the sagittal formula of a parabola.  Namely,
R=r^2/2s as g would then equal s.  For h=0 it fails.  More specifically,
the error increases as h gets smaller.  That is, as you use a smaller and
smaller drill bit to measure your radius or have a very fast mirror, you
get less and less from this approximation.  However, for the typical ATM
mirror, assuming that the drill bit stops an inch or so from the edge.
This approximation will produce results to about half a percent.

It is possible to include more terms in the approximation for g and still
solve for R.  In fact, using MathCad I typed in g to second order and
solved for R.  The resulting solution reduces the error under the same
conditions to 0.002 percent and, the resulting formula is as long as this
entire post.

If I'm asked I'll post it, but I think it misses the point, given that this
was intended to measure rough grinding depth.

Anthony