Re: ATM HYPERBOLA

[Stuart Field <field@lamar.Colostate.EDU>]

Let a = R/(e^2 - 1), where R = radius of curvature of center zone.  Then

    z(r) = a  (sqrt ( 1 + r^2/a^2 ) - 1).

If you expand this in a power series you get

    z(r) = r^2/(2R) - 1/(8 a^2 R) r^4 + ...

Note the familar first term, which is the same for any conic section
(or, indeed, for any smooth curve with paraxial radius of curvature R),
and exact for the parabola (e = 1).

Thus for practical purposes you cannot tell the difference between a
hyperbola, sphere, etc. simply by measuring the sag and the paraxial R.
For instance, for a hyperbola of e = 2 the second term in the expansion
above is less than 1% of the total sag for a 16" f/4 mirror.  Maybe
marginally detectable.

Cary Chleborad wrote:

> Anyone have a simple equation for determining the sagitta of a
> hyperbolic curve of a given ecentricity?
>

Stuart Field