Re: ATM between a parabola & a sphere: computation

[Guy Brandenburg <gfbranden@earthlink.net>]

Thanks.
Guy B

"Good, Donald" wrote:
> 
> 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.