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What exactly is parallax anyway?

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Topic: What exactly is parallax anyway?
Posted By: Chris Farris
Subject: What exactly is parallax anyway?
Date Posted: June/05/2004 at 09:41

Parallax is essentially an optical illusion.  Parallax presents itself as the apparent movement of the reticle, in relation to the target, when your eye moves off center of the sight picture (exit pupil) or in more extreme cases it appears as an out of focus image. It indicates that the scope is either out of focus or more specifically the image of the target is not occurring on the same focal plane as the reticle. Maximum parallax occurs when your eye is at the very edge of the sight picture (exit pupil). Even when parallax is adjusted for a designated distance, there is an inadvertent error at other distances.  Most brands of scopes that do not have a parallax adjustment are pre-set at the factory to be parallax free at or around 100 yards; rim fire and shotgun scopes are set at or around 50 yards.  Most scopes of 11x or more have a parallax adjustment because parallax worsens at higher magnifications.  Generally speaking parallax adjustment is not required for hunting situations and is primarily a feature used and desired by target shooters.  A 4x hunting scope focused for 150 yards has a maximum error of only 8/10ths of an inch at 500 yards.  At short distances, the parallax effect does not affect accuracy. Using the same 4x scope at 100 yards, the maximum error is less than 2/10ths of an inch. It is also good to remember that, as long you are sighting straight through the middle of the scope, or close to it, parallax will have virtually no effect on accuracy in a hunting situation. - RifleScope School


This is another way to think of it that maybe you can relate to.  You know when you are sitting in the passenger seat of a vehicle its hard to look at the speedometer and tell how fast you are going because your eye....the needle....and the mph number are not all three lined up.  So to you it looks like your going 35 when really you are going 55.  But the person behind the steering wheel has his eye..the needle and the mph all lined up straight in the same focal plane and gets a true reading.  Its not exactly the same thang but close enough for government work.

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Posted By: Chris Farris
Date Posted: June/09/2004 at 14:12

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.

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Posted By: Chris Farris
Date Posted: July/12/2004 at 10:21

This article was written on another site and is credited to Paul Coburn. I thought it made for interesting reading.

"I've answered questions about scope parallax about 300 times, and it's always a long drawn out thing, going several e-mails, and a few phone calls. It doesn't seem to make any difference how long the guy has been shooting, this one always keep screwing guys up.
OK... here goes (Whew, this is gonna be a long one).
There are several things that go on inside a scope, and in the eyes at the same time. Some of them workie against each other.
But some terminology first... and we'll leave out lenses that are there to correct some optical or color errors, but don't have anything to do with image forming.
We'll start at the front of it all, and work back.
1 - The "Object"... the thing that you are looking (shooting) at.
2 - The "Objective". The front lens is called the "Objective"... it forms the first image of the "object" we are looking at (that why they call it the Objective
It is the lens that "captures" all the light, that is solely responsible for the image quality of the scope... if it is poor, you can't fix the poor image later.
This lens is usually made of two different types of glasses (called "elements") sandwiched together, and is called an "Achromat".
The Achromat is fully color corrected for blue and green. The red wavelengths are partially corrected, but have what is called "residual color errors".
This is the normal type of objective used in shooting and spotting scopes. In quality, they can vary from badd, through sorta OK, to pretty damn good.
If one of the elements is made of an "ED" glass, or a "Fluorite" (CaF) glass, the two element lens can be very good to friggin' outstanding.
In some instances, objective lenses are made of three elements, and all three colors (blue, green, and red) are completely corrected. This type of lens is called an "Apochromat", and this is the finest lens that can be bought. The best of these can also have "ED" glass, or Fluorite as one of the elements.
3 - The "First image plane". The Objective focuses the light to make an image of the subject, just like a camera lens. This image is upside down, and right/left reversed. This is the first image plane, but NOT the "First image plane" that is talked about when shooters talk about reticles.
4 - The "Erector lens"... (if it is a group of lenses, it is called the "Erector cell"). Because the first image is upside down/wrong way around, we (as shooters) can't use it... so we flip it around with a simple optical group called the "erector cell". This cell gives us a new image that is right way around, called the second image plane. Moving this cell causes this second image plane to move... so micrometer spindles are put against the cell, to get elevation and windage adjustments.
5 - The "Second image plane". This is the second real image plane in the scope, and this is the image plane that shooters call the "First image plane" when talking about reticles. In a fixed power scope, or in a variable with a "First image plane reticle", the reticle would be placed in this image plane.
This is where Premier Reticle puts those magical "Gen II" reticles.
6 - The "Zoom group". In a variable scope with standard (non-magnifying) reticle, the zoom group of optics would follow #5. This group of lenses can change the size of the image plane in #5 and then form a new (third) image plane behind it.
7 - The "Third image plane" In variable power scopes, this is the plane that the reticle is placed in. By being here, it allows the image to change sizes, but the reticle to stay the same size. In the context of reticles, this is the image plane that is referred to as the "second image plane"
8 - The "Eyepiece". This optical group is like a jewelers loupe. Is is (or should be) a super fine magnifier. It's only job in the whole world, is to focus on the reticle.
Let me repeat that for those that live in Rio Linda...
It CANNOT adjust, or compensate for, or do anything else when things look bad in the scope, or when you can't hit the target... and you CANNOT use the eyepiece to try to correct for parallax. That is sheer folly at best, and raw stupidity at worst.
If you expect it to do anything else, then stop wasting your time with long-range shooting, cuz you are never gonna make it past mediocre... and take up golf!!
OK... now that you know what the insides are like... lets move on. We'll use the zoom scope for our examples. cuz if you can understand the zoom scope, then the fixed scope is a walk in the park.
In the scope that is set for infinity range, the object forms an image behind the objective (the first image plane)... the erector cell "sees" that image, and flips it over and makes it right way around in a NEW image plane (the Second image plane). The zoom group adjusts the size of this image plane, and makes a NEW image plane (the Third image plane) that is the desired size. There is a reticle placed in this last image plane, and the eyepiece focuses on the reticle AND the image at the same time.
When things are good, that's how the scope workie!
But... now the booger falls into the soup... IF the third image plane and the reticle are not exactly, (and I mean EX-ACT-LY) in the same place, then your eye cannot see them LOCKED together as one picture.
It sees them as two separate pictures, and the eye will look at each separately, and the eye can also look AROUND one to see the other.
Lenses are measured in metrics (aka Millimeters). Not because the Europeans wanted the metric system 20 years ago, but because optical strings and chains of lenses (like scopes) are really a string of numbers.
There are constant ratios of "this divided by that's" that give image sizes, "F-ratios", and image locations. It's so damn easy to do the engineering using a 10 based system that the optical guys were using the metric system way back in the 1800's.
The objective has a "Focal length"... this is the distance behind the lens that the first image plane falls when making an image if a subject that is at infinity (or very damn far away).
If the objective has a focal length of 100mm, then the image of that 1000 yd target is 100mm behind the lense.
But the problem with geometric optics (which is what we are dealing with here), is that they follow the laws of geometry... and optics make triangles like rabbits make babies.
AND... in an optical chain, when you change one thing, one angle, one ANYTHING, everything else follows along and changes BASED on the ratios involved at THAT stage.
If we take that same target, and move it to 100 yds, the image in the scope moves BACKWARDS, going further into the scope. Not by much, but it doesn't take much, cuz we're dealing with very small distances inside the scope, and very high magnifications.
How far the image moves back, and what it's new position is, is predictable by the mathematical ratios of the angles formed by the subject and the first image... OR (for us dummies that lost our slip sticks) by the ratio of the distances to the Target and the focal length, multiplied by the focal length. then ADDED to the focal length.
The target is at 100 yds (91440mm), the focal length of the objective is 100, so the displacement is 1/914 x 100, which means that the first image is now at ~100.1mm. Hmmm only .1mm, that doesn't seem like much.
Read the following paragraph twice...
In a 1x scope, 0.1mm would mean nothing... but this displacement is repeated throughout the chain, AND if any of the optical groups change the image ratio (aka image size), then the displacement (aka ERROR) is changed in direct proportion to the increase in magnification. So in a 3x scope, it would be .3mm, and in a 10x scope, it would

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Posted By: Chris Farris
Date Posted: September/24/2004 at 18:13

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
                  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!    (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 

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Posted By: Dale Clifford
Date Posted: September/25/2004 at 19:43
Scopes (and other none camera lenses) have parallex because all the light "rays" are parallel. They do not have an iris, thus no depth of field the longer or deeper the depth of field the better the focus.  (exept for a bunch produced by Burris about 10 years ago that worked very well, I don't know if they still carry them). If it were possible to select only those light rays containing the image against the crosshair without the additional surrounding rays discussions of parallex would be relegated to theorists. Chris's walk through of how to adjust the ocular is about the best I've ever seen and takes more time than doing. While theortically correct that you cannot correct parallex with adjustments to the ocular, you can "trick" it especially at short distances and this again depends on the power of the scope, as stated earlier. This process is a result in other random errors in the scope system, (width of reticle,compounding problems built into the erector systems of variables etc.) In most practical shooting situations the error correction without parallex correction (depending on power again) is less than the s.d. of spread on the group size the gun can shoot at that range using the power under considerations. This arguement and the ones presented above are the single "best" arguements for fixed power scopes. (especially when price in concerned). Variable scopes are a result of consumer demand-not an optimization in the shooting system.

Posted By: Chris Farris
Date Posted: October/30/2006 at 11:58

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.

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Posted By: Chris Farris
Date Posted: March/24/2011 at 14:38 -

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