Re: [ATM] parabolic versus spherical volume of glass

["vladimir sacek" <v959@msn.com>]

Don,

I find the visual reference of particular parabolas vs. initial sphere
helpful. So much so, that I'll risk boring some people to death by
going over it. 

A sphere, being undercorrected, has the edge focus shorter 
than the center. The foci separation (longitudinal aberration) is given 
by D/32F, D being the diameter and F the F#. There is an infinite number 
of parabolas that focus between these two foci, but only four of 
them are sufficient to illustarte the sphere/parabola relationship.

#1 is a parabola that focuses at the paraxial focus. It's radius is
equal to that of the sphere. Being flatter, it is shallower
than the sphere, with the difference in depth increasing towards the
edge, where it reaches its max of D/1024F^3. It can be arrived to by 
taking as much of the glass off the edge, diminishing to near-zero 
towards center.

#2 is a parabola that focuses at the best focus, in the middle between
the paraxial and marginal focus. Obviously, it is stronger (more bent) 
than #1, and it is in fact of equal depth as the initial sphere.
Being flatter, it fits entirely within the sphere, with the maximum 
deviation from it at the 0.707 zone. It can be arrived to by taking equal 
depth of glass from both, edge and center, diminishing gradually to 
near-zero taken off the 70% zone. Depth of glass taken from edge and center
is 1/4 of the depth taken from the edge of the #1 parabola, or ~D/4000F^3. 
Total amount of glass to remove is over four times less than #1..

#3 parabola focuses at the circle of least confusion, at 1/4 longitudinal 
aberration from the marginal focus. It is stronger yet than #2, and deeper
than the initial sphere. It can be arrived to by taking about six times more
glass from the center than from the edge, diminishing to near zero at the 
87% zone. Depth of glass taken off the edge is about three timess smaller
than with the #2 (or~D/13,000F^3). Total amount of glass removed is about 
50% greater.

#4 focuses at the marginal focus, closest to the mirror. It is thus stronger 
than #3, and deeper from the sphere by as much as #1 is shallower. It can
be arrived to by deepening the center most, by ~D/1000F^3, and gradually 
less outwards, to near-zero at the edge. It requires nearly 20% less glass 
removed than #1.

Parabolas #1 and #4 are more likely to have edge problems; the former due to
most of work concentrated around the edge, the latter due to too little work 
around the edge. They also require significantly more glass removed. The choice 
parabolas seem to be those between #2 and #3, with most of glass removed 
from the center but also a small to moderate amount from the edge. They seem 
to have best chances to end up with a quality surface, in the least amount of
time. 

As a side note, I went through those equations for Paul-Baker I've posted, and
noticed a typo: primary conic is -1, not 0 (wishful thinking). Sorry for the 
confusion.

Greetings,

Vlad



  After reading your post, I realized my statement about positive 
  sag curve values meaning adding glass was incorrect.  It actually
  means that the maximun curve value is a zero reference level from
  which glass can be removed in figuring (both inward and outward).
  I have no idea what figuring strokes would be used to achieve it.
  Maybe something similar to the technique used to remove a 70% 
  zone valley.

  I have added a column to my sagitta.xls spreadsheet to calculate
  the volumn of glass removed to cover the case when parts (or all)
  of the sag curve is positive.  See:
  http://www.atmlist.net/contrib/donald-dot-good-at-comcast-dot-net/index.htm<http://www.atmlist.net/contrib/donald-dot-good-at-comcast-dot-net/index.htm>

  Clear skies,
  Don


  > -----Original Message-----
  > From: atm-bounces@atmlist.net<mailto:atm-bounces@atmlist.net> 
  > [mailto:atm-bounces@atmlist.net] On Behalf Of vladimir sacek
  > Sent: Tuesday, January 04, 2005 11:26 AM
  > To: atm
  > Subject: Re: [ATM] parabolic versus spherical volume of glass
  > 
  > Still not done. Guy did me good with his question, because it 
  > heleped me clear some misconceptions. A 70% zone doesn't 
  > focus at the circle of least confusion - as I unjustly 
  > accused the literature of claiming - but at the best focus of 
  > the spherical mirror. It is so obvious, wonder how I haven't 
  > noticed it. Guess my brain was in one of its dumb modes.
  > 
  > After putting things together, I crunched some numbers, which 
  > showed that least amount of glass to remove is really for the 
  > parabola focusing at the best focus, where similar
  > (greatest) depth of glass is taken from both center and edge, 
  > and 70% zone is worked the least the least (~0.0025 ci for 
  > 16.5" f/5 mirror). Working center somewhat more than the 
  > edge, and ~87% zone the least, requires somewhat more glass 
  > removed (~.0038 ci ), but has the advantage of less activity 
  > around the edge. Deepening from the center out requires
  > ~0.009 ci removed, and from the edge in (least desirable)
  > ~0.011 ci. (the former parabola focuses at the marginal focus 
  > of the initial sphere, the latter at the paraxial).
  > 
  > Why do I have to spend time trying to figure out things that 
  > are already known?
  > 
  > Vlad
  > 

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