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    Re: Fw: Re: A simple three-body fix puzzle
    From: Frank Reed
    Date: 2010 Dec 9, 17:38 -0800

    I wrote:
    "I'm also attaching the scan of the relevant paragraph. Note that in the
    last line, L and B with subscripts refer to longitude and latitude, but all
    we need here is the difference in these in miles which is given by
    (AE-BD)/G and (CD-BE)/G".

    George H, you wrote:
    "No, not in miles."

    Yes. IN MILES. If the intercepts are in miles, then (AE-BD)/G and (CD-BE)/G are in MILES (A,B,C, and G are all dimensionless --just numbers-- while E and D have the same units as the intercepts). Now, there may be some navigators somewhere on the face of the Earth who normally give their intercepts in degrees, but I have not met one. To you navigators out there who DO quote intercepts in degrees, I am confident that you can make that oh-so-terribly-difficult unit conversion on your own. For everyone else, my previous statement was correct.

    George, you went on at some length about the "iterative" aspect of these equations, apparently imagining that this is a fundamental property of them, and then you described using them to find a two-body fix, possibly in another hemisphere. This is like using a wrench to hammer a nail. These equations are intended to replace the plotting step on a large-scale chart that we're all familiar with: drawing LOPs at their respective azimuths based on their intercepts from a point near our estimated position and then finding the point where they all cross (as nearly as possible), but the equations do so in a statistically sound way in a least squares sense. And EXACTLY like the plotting step, these equations are based on a simplifying geometric assumption of a flat Earth and straight LOPs. The idea that these equations have to be iterated when the new position is more than a dozen or two dozen miles from the assumed position is no more remarkable than saying that one should re-work a standard LOP plot under similar circumstances. And in fact, if you're using this system of analysis properly, you would almost never find yourself in a position where you need to iterate. Sure, you can coax these equations into the more academic task of finding a fix when the original position is completely unknown, which does require multiple steps of iteration, but that's not what they're for. ...Get a hammer when you need to hammer a nail.

    Finally, none of this business about iteration has any bearing whatsoever on the topic being discussed previously in this thread nor does it invalidate or even modify in any useful way anything that I wrote previously.


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