Re: ATM 21" F/6: response about sagitta equations

["Richard Schwartz" <richas@idt.net>]

Dominic-Luc Webb molmed wrote:

> S(r) = R - (R^2-r^2)^0.5   eq (1)
>
>
> Richard Schwarz claimed that:
>
> S = r^2 / 2*R    eq (2)
>
> is exact, but for a parabola. On page 15, Neale
> Howard claims that this same equation is an
> approximation for a circle, and not a parabola.

Why are we doing this?   The equation I gave is exactly right for a
parabola, and is also an approximation for a circle with the same value of
R.   You can see this by doing a Taylor series expansion of (1).  Will
everybody please check my calculus, and don't forget this formula?  (It was
hard to get this right, sitting in the coffee shop with pretty college girls
pestering me constantly!)

(3)   S(r) = r^2/2*R   +  r^4/8*R^3   +  r^6/16*R^5 + 5*r^8/128*R^7 + ....

> Further, in the footnote at the bottom of the
> page, he goes on to say that this equation will
> not suffice for focal ratios above F/5 and that
> eq(1) must be used.

Equation (2) is exact for all parabolas, no matter the focal ratio.  As far
as I know, most telescope mirrors, no matter the focal ratio, are parabolas,
not circles.   Equation (1) is exact for circles, but the focal ratio is not
allowed to be less than f/.5, or you get unreal numbers.     Equation (3)
above for a circle is as exact as you need to be, depending on how many
terms you keep.   Equation (3) is valid wherever equation (1), from which it
is derrived, is valid.   You know how exact it is by the value of the last
term you evaluate.

For my favorite mirror, a 200" f/3.33, if it were a sphere, I get the
following terms:

S(100") = 3.75375375 + .00528928 + .000014095 + .0000000525 + ...

Note that this last term is about 1/380 wave, probably beyond our ability to
measure, fabricate, or care about.  Note that the first term is plenty to
worry about, and is probably beyond the ability of pitch laps to correct
within the span of a normal human lifetime.  The correction was probably
worried about and ground in starting with with #400 grit.

So, we now get to the underlying problem... what is the difference between a
parabola and a circle?  That is the difference between formulas (2) and (3),
which I leave as an exercise for the student to calculate.

> that my original claim is correct and that eq(2)
> cannot be for a parabola, or, Neale Howard
> may be in error. Perhaps I missed something?

Use the circle formula (1) or (3) for a circle, and the parabola formula (2)
for a parabola.  Both are exactly right.   If you use the wrong formula, you
get an approximate answer that for many practical applications will be as
close as you need to be.   To find out how wrong, use the power series
above.

I don't want to ridicule any stupidity on the ATM list; after all, is light
a wave, or is it a particle?   How do you prove it?

. . . Richard