From: bercov@bevsun.bev.lbl.gov (John Bercovitz)
Subject: scope parallax, was Re: Colt M16A2 vs. Colt AR-15 Delta HBAR
Organization: Lawrence Berkeley Laboratory

There may be a modest amount of confusion out there on the subject of scope
parallax.  Parallax problems result from the image from the objective not being
coincident with the crosshairs.  (On high magnifications scopes, the objective 
is the big end of the scope; vice-versa for low power scopes; in either case
it's the guzin end.)  If the image is not coplanar with the crosshairs 
(that is the image is either in front of or behind the crosshairs), then
putting your eye at different points behind the ocular causes the crosshairs
to appear to be at different points on the target.  (The ocular is the guzout
end of the scope.)  In fact, this is the basis of a test for parallax problems:
Set your scoped rifle on sand bags.  Align the scope with the center of the 
target.  Without touching the rifle, move your eye around behind the scope.
Do the crosshairs appear to move on the target?  If they do, the parallax is
not set for the range of the target you are using.

So which way do we move the objective to correct parallax?  First hold up
the index finger of one hand in front of the palm of the other hand.  (You
don't have to actually DO it, this a thought experiment.)  Let the index
finger represent the crosshairs and the palm represent the image plane.
If you move your head to the left, the finger moves to the right against the
palm.  So if your crosshairs move to the right on the target's image when
you move your head to the left, the image plane must be further away than the
crosshairs.  What's a mother to do?  Why pull the image plane in a little
by screwing the objective bell in so that the objective moves closer to you,
of course.  In this set up, the image is essentially tied to the objective
so moving the objective 0.1 mm moves the image 0.1 mm.  And no, the ocular
doesn't change this scenario any more than putting a weak loupe to your eye
would change the sense of the thought experiment using index finger and palm.

As long as we're on the subject of scopes, I might as well mention focussing
the ocular or eyepiece (same thing).  The goal here is to focus the ocular,
which is really just a magnifying glass, on the _crosshairs_ which are located
just ahead of the ocular.  To avoid the distraction of the objective's image,
you can cover the objective with something translucent like maybe a sheet of
Kleenex.  Screw the ocular out, away from the main body of the scope until the
crosshairs go out of focus.  Now screw it in until the crosshairs are just in
focus and then turn it in a little bit more.  This puts the crosshairs slightly
nearer than infinity as far as your eyes can tell.  Your eyes will appreciate 
not having to strain to focus on the crosshairs, especially if they're old eyes
like mine.  Even if you have young eyes, a long day of varmint shooting will 
strain your eyes if you've focussed your ocular by reversing the sense of the 
above procedure. 

After you have focussed your ocular, you can set your parallax by the procedure
delineated in the above paragraphs.  This is quite often a more accurate way of 
setting parallax than setting by the yardage lines inscribed on the objective 
bell (on many brands those lines are approximate at best).  

Warning!  Snoozer follows!

Now can we calculate?  Oh, goodie!  On a short scope, the objective's focal
length must be around 0.1 m considering that there is an erector lens in that
tube also.  The formula for the distances from a lens of the object and the 
image of that lens is:
                          O^-1 + I^-1 = F^-1
where:
                 O = distance from object to lens
                 I = distance from image to lens
                 F = focal length of lens

What I'd like to know is how far we'd have to bring the objective lens in
if we shift the parallax correction from 50 m to 100 m.  Moving the objective
lens relative to the scope body makes no essential change in the value of
the variable, O.  So how far is the image from the lens when the target is at
50 m?   100 m?   150 m?

        I(50) = [(F^-1)- (O^-1)]^-1 = [(.1^-1)-(50^-1)]^-1 = .1002 m
        I(100) = [(.1^-1)-(100^-1)]^-1 = .1001 m
        I(150) = [(.1^-1)-(150^-1)]^-1 = .10007 m

We can now see that we're talking very small parallax correction movements here
and that furthermore, the corrective movement required for an increment in 
target distance decreases rapidly as the distance to the target increases.
So the answer to my question is, if you move the target from a 50 m distance to
a 100 m distance, the objective must be moved .1002-.1001= .0001 m to correct
the parallax.  In Marekin terms, this is .004".  That sounds about right to
me considering that the graduations on an objective bell are fairly close 
together and the objective bell's thread is very fine.  This also explains
why it is difficult for the scope manufacturer to put the parallax marks on
the bell in exactly the right place.  All eyes are closed?  Have a nice sleep!

       JHBercovitz@lbl.gov    (John Bercovitz)

From: bercov@bevsun.bev.lbl.gov (John Bercovitz)
Subject: Re: Parallax adjustments on scopes(clarifications & corrections)
Organization: Lawrence Berkeley Laboratory

In article <39764@mimsy.umd.edu> gmk@falstaff.tmc.edu (Columbo Kotzar) writes:

##The above definition of parallax is correct for rifle scopes. The way parallax
##errors occur is that the primary image -I am used to dealing with real objects
##not virtual objects, silly me- is brought into focus on a plane that is not
##coincident with the plane of the reticle. When that occurs moving your eye

The images in a scope are real, not virtual, so you got it made!  8-)

##across the field of view results in the crosshairs moving relative to your
##target. The way this is corrected is by moving the objective element(s) to
##focus the image of your target on the same plane as the reticle. The movable
##objective element(s) actually do two things: first is focus the image of the
##object and second is fine tune where the focused image lies in the body of
##the scope.

I know you know the following, Geoff, but I think the above may be misread.
You don't want to focus the scope with the objective.  You focus the
reticle with the ocular and then correct parallax with the objective.
Certainly if you have a scope adjusted correctly and your eyes don't have
much accommodation left and you fool with the objective, the image will go
out of focus, but that's a side effect.

##How much error are we talking about? I don't know at the moment but I have
##heard that 1/4 inch figure for scopes set for 100 yards when used at 50 and
##have seen about that amount when using one of the LER pistol scopes at 100 yds

#I hate to drag this out much further but there was one point that I overlooked
#and wanted to include. The magnitude of the error caused by parallax is a
#function of the scope magnification, at least it appears this way. The 1/4 inch
#number given above was for a 4X scope. As the scope magnification increases
#beyond about 9X parallax adjustment becomes important, so if you need 10X and
#greater magnifications you might want to look into a model that allows cor-
#recting for parallax (no pun intended). If you really only need 9X and less,
#you are probably shooting at something big enough that parallax errors are not
#important.

I tried to do a little calculating on this and got stumped at the point of 
figuring the effect of axial magnification of the ocular so I called Leupold and
got their answer man, one Merwyn Webb.  As I suspected, axial  magnification 
doesn't really play a part in this.  So it's all really rather straight forward:
According to Webb, regardless of scope magnification, if the objective's image 
is .001" in front of or behind the reticle, the parallax error is 1" at 100 
yards for the condition of the eye being at the extreme edge of the exit pupil, 
at least to the first order.  (It also depends on the diameter of the exit pupil
inasmuch as this sets the latitude you have in placement of your eye.)  Since 
this is an angular problem, 1" at 100 yards is equivalent to 2" at 200 yards.  
The reason it doesn't bother you in a low power scope is that this magnitude of 
error is too small to see in a low power scope.  I asked him if the focal 
length of the objective was around 0.1 m as I speculated in an earlier post and 
he said it was around that but it varied since the objective and erector often 
work together to set the focal length (ie, the erector often is not just a pure 
erector).  Also, scopes designed for different purposes have different focal
length objectives.  If my figure of 0.1 m is correct, the image to reticle 
distance is .0001 m or .004" for a scope used at 50 but adjusted for 100 yards
(or vice-versa), as shown in an earlier post.  This would correspond to a 4" 
error at 100 yards or a 2" error at 50 yards if Mr. Webb is also correct.  This 
sounds slightly high to me.  I guess I'll just have to try this experiment and
see what happens.

#Even if they are, slow down and place your eye along the scope axis
#and the error will go to zero.

Good advice.  It's really not much harder to get your sighting eye on the
axis of a scope than it is to get it on the axis of a peep sight.

       JHBercovitz@lbl.gov    (John Bercovitz)

From: bercov@csg.lbl.gov (John Bercovitz)
Subject: Re: Dot-type scopes?
Organization: Lawrence Berkeley Laboratory, California

In article <CAvouI.L8o@fc.hp.com> bartb@hpfcla.fc.hp.com (Bart Bobbitt) writes:

#Mickey Boyd (boyd@gauss.math.fsu.edu) wrote:

#> I agree.  The big advantage (IMHO) of dot sights is that they offer true
#> 1 power sighting,. . . . with no parallex.

#I don't think this is true.  As the objective lens group does focus the
#target's image at the reticule plane, parallax can occur.  The reason is,
#......

I think one reason that it's common lore that a 1X scope has no parallax
is that the parallax is not magnified so it's not readily discernable.
I've seen zero parallax claimed for 1X scopes in just about every place
possible, books, magazines, advertizements.  Eyes with good acuity will
show it, though, more so at close ranges.

#I once had a Weaver K1; an excellent zero-power scope.

Ain't that one-power?  Seems like zero power would provide you with
a point rather than a field of view.

#.......I unscrewed the objective lens out a ways to focus
#the scope at 25 yards and the parallax went away for all practical
#purposes.....

This is getting very picky but I think it might be less confusing to say
that you unscrewed the objective to make the objective's image coincide
with the reticle.  Focussing would then be done with the ocular (eyepiece).
On the other hand, maybe this is a more confusing way of saying it!  8-)

John Bercovitz     (JHBercovitz@lbl.gov)

From: Louis Boyd <boyd@pegasus.la.asu.edu>
Subject: Re: Leupold varix-iii Questions.
Date: 24 Jan 1997
Newsgroups: rec.guns

bartbob@aol.com (Bartbob) wrote:
#About these words:
#<<< The maximum error possible due to parallax is less than 1/2 the
#diameter of the objective.  To get that you have to put your eye at the
#extreme edge of the exit pupil AND be at a distance either much greater or
#much less than the range the scope is adjusted to.>>>
#
#Having measured parallax error in scopes focused at 50 and 100 yards for
#ranges of 1000 yards and finding the error to be a few feet, there are big
#errors in the background of the above words.
#
Bart, you are right that I was in incorrect on the magnitude of the parallax
error. The error I gave would be the maximum error as the bullet passed
through the focus distance, not the error at the target.   Of course,
no one cares about that error, only at the target :-)
The actual maximum parallax error at the target is:

max error =  0.5 * objective diameter * (range to target-range to focus)
             ____________________________________________________
                              range to focus

Any units will do as long as the units for the two ranges are the same and the
units for the objective diameter and the error are the same.

For a 2" objective on a scope focused at 50 yards shooting at 1000 yards
the maximum error would be 19 inches.

#It doesn't matter whether the objective lens is 1mm or 1 foot
#or 1 yard in diameter; the focus points of the two targets are 9/10ths
#inch apart along the scope's optical axis because of lens group focal
#lengths and target distance.

Here you're partially correct.  The magnitude of the error is independent
of objective diameter for a given distance that the eye is offset.  However,
as the objective size is increased, the exit pupil grows larger linearly
with objective diameter.  This allows the eye to be positioned further
(perpendicular) from the optical axis before the reticle and target are
no longer visible, thus the >maximum< error will be greater with a
larger objective and it will increase linearly with objective diameter.

The original poster was inquiring whether he should buy a fixed or AO
Leupold scope.  The factory adjustment for the fixed scopes he mentioned is
150 yards.   The scopes he mentioned are available with either 40 or 50 mm
objectives, so the error at a thousand yards could be:

for 40 mm objective : 4.46 inches
for 50 mm objective : 5.58 inches

Thats's a lot bigger than the 1" I indicated in my original post but it's
a fair amount smaller than what you have stated. Please correct me if I'm
still wrong. You caught my mistake though. Thanks!