RE: ATM between a parabola & a sphere: computation

["Good, Donald" <dgood@aha.org>]

Just a comment on your circle eq to start, you took the long way around,
but got there:
>From (x-0)^2 + (y-R)^2 = R^2, get
(y-R)^2 = R^2 -x^2 then
y-R = +/- sqrt(R^2 -x^2) and finally
y = R +/- sqrt(R^2 -x^2)

Your final eq  (absval ( x^2 / (2*R) - R + sqrt (R^2 - x^2) ) is correct
for the difference in a static sence.  But during figuring, the glass is
removed, mostly from the center, deepening the curve into a parabola.
Some also comes off the edge, flattening out the circle just a bit.  So
what is happening in the equation is that the vertex moves down just a bit
(the R near the center gets smaller) and R near the edge gets a little bit
larger.  This is just what is necessary to correct the spherical aberation
where the central area of a sphere focuses too far and the edge area focuses

too near.

To calculate the approximate glass removed most accurately, write the
equation of your final parabola.  Then find the equation of the circle that
is tangent to the parabola at about 71% (1/sqrt(2)) of the radius of the
mirror.  The circle will be above the parabola at both the center and the
edge.  This is near the minimum glass to be removed.