What exactly is parallax anyway?

Here is what Webster has to say:

par·al·lax  n. - An apparent change in the direction of an object, caused by a change in observational position that provides a new line of sight.  The apparent displacement of an object caused by a change in the position from which it is viewed.  Parallax of the cross wires (of an optical instrument), their apparent displacement when the eye changes its position, caused by their not being exactly in the focus of the object glass.  Binocular parallax, the apparent difference in position of an object as seen separately by one eye, and then by the other, the head remaining unmoved.

Here is another way to understand what parallax is and what causes it.

With one eye closed hold your thumb out in front of your open eye and put your thumb on top of a distant object.  Now close the eye you are looking through and open the other eye while you hold your thumb steady on target.  Your target is now visible and your thumb will have shifted to the left or right.  The target did not move nor did your thumb but somehow they are not on top of each other any longer because the observation point changed.

Interesting information I stumbled across on the Internet

Subject: scope parallax

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)

Subject: Re: Parallax adjustments on scopes(clarifications & corrections)
Organization: Lawrence Berkeley Laboratory

In article <39764(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.

Subject: Red Dot-type scopes?
Organization: Lawrence Berkeley Laboratory, California

In article <(Bart Bobbitt) writes:

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-)

How Parallax Applies To Shooters

by Hugh Birnbaum

Parallax is an interesting phenomenon we experience continuously while viewing the world around us.  It helps us navigate the three-dimensional mazes of daily life.  In practical terms, parallax is the apparent change in spatial relationships between elements in a three-dimensional scene that occurs when we alter our viewing position.  Move slightly, and stationary objects located at different distances in the scene seem to shift their positions relative to each other.  Parallax may become troublesome, however, when we encounter it in optical sights.

When we look through a scope and see the reticle neatly planted against the target, we are likely to interpret what we see as a single "picture," something akin to a photograph, when what we actually see are two pictures that are superimposed.  The forward, or objective, lens system of the scope forms an image of the target area, and the rear, or ocular, lens system forums an image of the scope's reticle.  Ideally the two images coincide at the same plane within the scope as though they were printed on an invisible plate.  The result is that you cannot make the reticle image shift relative to the target image, even if you change your eye position from dead center to the far edge of the eyepiece.

We do not live in an ideal world.  Most general-purpose scopes are focused at the factory for a particular target distance, typically about 100 to 150 yards.  A target at the prefocused distance will look clear and sharp through the scope, and its image will fall on the same plane as the image of the reticle.  There will be no visible parallax discrepancy regardless of your eye position, and you may reasonably expect your bullet to strike the target where you set the reticle.

If the target is significantly nearer or farther than the scope's optimal distance, its image will be less sharp, although not always obviously so, and will formed within the scope a bit ahead of or behind the image of the reticle.  The images will be separated in depth.  If your eye is well centered with respect to the scope's eyepiece, you will view along the scope's optical axis and the separation in depth of the reticle and target images will have no negative effect on the outcome of the shot.  If your eye drifts off center, though, a slight apparent shift will occur between the relative positions of the reticle and the target images.  You are now likely to move the firearm to place the image of the reticle where you first established it on the target.  You have parallax error in aiming, and your shot will impact slightly away from where you expect it to land.  Fortunately, parallax error is rarely great enough to spoil an otherwise well-executed shot in the field.  Furthermore, the human eye has a proclivity for centering itself well with respect to apertures through which it is viewing, such as a scope eyepiece, so extra care when mounting the firearm will help, too.

Parallax error is a more serious concern for target, benchrest, and varmint shooters, who place a premium on precise shot placement.  Riflescopes designed for these demanding application sallow you to adjust the objective lens for exact focus from 40 yards (some airgun scopes focus down to 10 meters).  The focus control may be a calibrated collar on the scope's objective bell or a rotary control on the turret saddle opposite the windage knob.  With either type, take distance scales with a large grain of salt until verified, because they are sometimes surprisingly fanciful, even on expensive instruments.

You can check a scope easily for exact focus and freedom from parallax.  With a fixed-focus model, set a target at the distance the manufacturer lists as the factory standard.  Immobilize the scope or scoped firearm in a steady rest or sandbag array with the reticle centered on the target.  Without touching the scope or firearm, move your aiming eye slowly from the center of the eyepiece to the edge while observing the reticle's position on the target.  If the reticle seems glued to the target, with no shift in position, the scope properly focused and there is no parallax at that distance.  An inch or so of parallax error is tolerable in a general purpose scope.  If there's much more than that, consider having the scope serviced.

With an adjustable-focus scope, check the focusing scale for accuracy using targets at known distances.  If it's slightly off, tweak the focus control until the target looks sharpest and there is no visible parallax error.  Repeat the procedure for each relevant target distance.  As you go, remark the scale with small dots of paint or nail polish, or a stick a marked strip of tape over the factory calibration.  If the scale is way off, send the scope back for a proper fix.

When performing focus tests with a scope that is already mounted on a firearm, do so at a shooting range or other safe venue.  Neighbors may find it unsettling to see your rifle poking out of the living room window while you focus on a lamppost that's a convenient 100 yards down the street.