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Parallax error in scopes and nonsense formulas I see on the net.

Discussion in 'Shooting Gear and Storage' started by Mr transformer, Mar 14, 2013.

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  1. Mr transformer

    Mr transformer Member

    Jun 21, 2011
    This is more of a rant to a degree.

    When ever I have looked for any specifications for normal parallax error in certain scope models, I always end up back at the formula I have seen all across the net. The one based on objective lenses size. If you have ever looked for parallax error figures, then you have most likely seen it too.

    That being...
    max error = 0.5 * objective diameter * (range to target - range to focus) / range to focus

    I went searching again for some data in regard to parallax error. All I could find was the same stupid formula, no mater where I looked. This last time was the straw that broke the camel’s back. So I am going to post a simple explanation of why that formula is nonsensical. It does not hold true in a simple optical system, let alone a complex one like a modern hunting scope. The thing that determines the parallax error is the design of the optical path, and lens system inside the scope, It has nothing to do with the primary objective lens.

    Down to basics.
    The cause of parallax error.
    The lens system in the scope focuses the image at a specific point in the scope. That point is called the focal plane. If the scope is set for a zero parallax error at 100M. When the scope is looking at an image at 100M, the image will be focused at a focal plane that falls at the exact same plane as the reticle. When the image moves closer, or further away, then the focal plane that the image falls on in the scope will move. When the focal plane of the image is separated from the reticle plane, then it will cause a shift in the location of the reticle in reference to the image when you change the location from which you are viewing it.

    It is like an image on a flat piece of paper. You set a pointer right on the paper, and you look at it from different angles. The pointer will always be pointing at the same spot on the paper. If you lift the pointer up where it is a inch off the paper, and then you look at it from different angles, then the apparent position of the pointer will change in reference to the image on the paper.

    Lets look at why the objective lens rule is nonsensical even in regard to a scope using a simple, single lens system.

    Example A,
    Lets say we have a scope using a 50mm lens. The edge diffraction rating is 45 degrees. That gives a focal distance of 25mm for an object located at infinity. The focal plane of an image will be 25 mm from the center of the lens. If you put a reticle at that point (25mm), the reticle will appear to be stationary, no mater what angle you look at it. There will be no parallax error. The total viewable optical cone from that setup will be 45 degrees. You will be able to get off center 45 degrees, and still see the image in reference to the reticle.

    As the object gets closer to the lens (less than infinity) then the focal plane will get further away from the lens. It will no longer be at the same plane as the reticle. When you view it at an angle, then you will see an apparent shift in regard to the image, and the reticle. I don’t want to go into exact figures, so I will use relative units of measurement. Lets say that the reference object moves to point X, which will cause the focal plane to move one unit out from the reticle. At a maximum viewable angle of 45 degrees, the reticle will move approximately 1 inch in reference to the image.

    Example B,
    Now, take the same 50mm lens, lets change the lens to crank the focal distance out to 50mm. that would be a lens with an edge refraction angle of around 26.6 degrees. Lets put the reticle at 50mm to yield zero parallax at infinity. Now, lets move object back to point X. since the focal distance is twice example A then the focal plane will move 2 relative units away from the reticle. Since the viewable optical cone is only 26.6 degrees then the viewable offset is about the same as the first example….BUT!!!!!! since the image is at twice the focal distance, then the image at the focal plane is twice the size. That will yield a parallax distance of only half an inch in regard to the target.

    So you have two examples with a simple lens system using the same objective lens size, with same object distances, but based on optical properties, and element spacing, you will get two different parallax errors.

    A modern scope is a compound/complex lens system and the parallax error can be changed by many factors in the optical path.

    The parallax error is strictly based on the internal optics of the scope. The objective lens size can be changed an order of magnitude and have almost no effect on the parallax error.

    That is why I wish people would ditch that stupid objective lens formula and actually give the normal parallax error for different models of scopes, instead of referring them to a nonsensical formula just to keep them from continuing to ask the question.

    I have no idea of exactly how the formula got started. It may have been dreamed up by someone to get people to stop asking him questions. What you tell someone doesn’t have to be true, they just have to think it’s true, so they will be satisfied, and leave you the heck alone.

    If you want, I can post illustrations to show the actual geometry of the interaction, but that would be beside the point. I just have gotten sick and tired of seeing the dang formula running around the web. So I thought I would throw my verbal rock at it out of disgust, if nothing else.

    Have a good day.
  2. Mr transformer

    Mr transformer Member

    Jun 21, 2011
    Here are a couple simple pictures I made to illustrate the point.

    The first is a simple single lens scope that shows how a change in internal optics can have an effect on parallax without a change in objective lens size.

    The second picture shows the actual factors that affect the parallax in a modern scope. The size of the lens that illuminates the reticle (the movable zoom lens) and the distance from the lens to the reticle is the primary limiting factors in maximum viewing offset. That offset verses focal plane offset by a close object is what causes parallax. That is determined by internal construction, and the objective lens size has no effect on it.

    And that parallax in reference to the scale of the image on that focal plane will determine the total actual parallax shift in inches.

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    Last edited: Mar 15, 2013
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