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

[Dwight Elvey <elvey@hal.com>]

Dominic-Luc Webb molmed <Dominic.Luc-Webb@molmed.ki.se> wrote:
> 
> 
> Richard Schwarz claimed that:
> 
> S = r^2 / 2*R    eq (2)
> 
> is exact, but for a parabola. On page 15, Neale 

Hi
 Here is how the equation 2 can be derived.
First we have to agree on how one defines a parabola.
For us telescope people the following would be the important
definition since it shows how the wave properties of this
curve make a good telescope.

 If one draws a line perpendicular to the axis of the
curve at the focal point and then chooses any point along
that line and draws a line parallel to the axis to the
surface, the sum of the distance between the first line
to the intersection of the curve and the line from the focal
point to that intersection will be equal to two times the
focal length. ( watch signs if the curve intersection is
on the other side from the axis intersection but this still
hold true ).

Using this as the basis, one can quickly see that the distance
from the perpendicular to the surface is the focal length
minus the sagitta at that radius and that the distance between
the intersection and the focal point must then be the focal length
plus the sagitta. Using the formula for the sum of the squares
of the sides of a right triangle, you get:

 r^2 + ( F-S )^2 = ( F+S )^2

Expanding this equation and simplifying you get:

 r^2 / 4F = S

Since R is defined as the center of curvature for the vary
center spot of the mirror, 2F = R. Simple substitution will
get:

 r^2 / 2R = S

 The reason it is stated that fast mirrors may not follow this
is that early on, while grinding, the sagitta is for a sphere,
since that is what you hope to have.
 Now that we have that clear, I don't think that either formula
should be used for more than a rough level not to exceed
while rough grinding. This is because you are never really sure
if you've properly brought the curve out to the edge while rough
grinding. I highly recommend using the sun focal length
measurement instead when you are getting close. This shows you
whether you need to keep the same curve and bring it out to
the edge when compared to the sagitta value measured from the
edge. Failure to check this can result in to deep a grind when
finally brough out to the edge. Using a spherical gage is another
method if the gage diameter is small compared to the mirror.
IMHO
Dwight