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    Re: Symmedian point -earliest reference in navigation
    From: Bill Lionheart
    Date: 2018 Oct 16, 09:18 +0100

    Thanks Brad.
    1) We don't. As in many experimental techniques if you calibrate
    everything you can, practice making the measurement enough, and
    average over a number of measurements the BEST you can expect at the
    end is independent, identically distributed, zero mean and
    approximately normal. Also this is the only assumption that makes it
    easy to calculate the most probable position, pretty much anything
    more sophisticated and you will need a computer.
    I am sure more experienced CelNav practitioners will come in on this,
    but of course if you take a series of sights of the same body you also
    get some information about the errors.
    2) If there are no immediate dangers yes I recommend the symmedian
    point - however the time spent finding it might be better spent taking
    a few more sights! As to an update on the corner nearest danger
    principle, I have something to say on that, but not finished writing
    it yet. The take home message is that the symmedian is close to the
    short edge!
    I agree these are good things to consider. I generally give this talk
    to 6th form students who have come for an interview at the University
    of Manchester. It is just a popular talk to entertain and enthuse
    them, and if they develop an interest in geometry or navigation that
    is a bonus. I would quite enjoy talking about this to an audience who
    care about navigation but I have not done that yet.
    On Mon, 15 Oct 2018 at 20:19, Brad Morris  wrote:
    > Bill, you wrote
    > I have searched through the NavList archives, I followed most of the 
    references in David Burch's helpful blog post 
    > From David Burch Navigation
    > Begin quote
    > "Practicing navigators have tended to choose the best position within the 
    triangle of intersecting LOPs (cocked hat) as some central value of their 
    choice, based on their experience and the actual sights at hand. In most 
    cases this is an adequate solution, but in rare cases ...
    > ...It can be shown that if the standard deviations of the sights are all the 
    same (no one LOP better than another), and there is no systematic error that 
    applies to all of them, then the most likely position is located at what is 
    called the symmedian point"
    > End quote, emphasis added.
    > -----
    > By following this advice, we may "improve" our fix, but only under certain 
    circumstances.  At the end of the geometric construction (or the matrix 
    manipulations, if so inclined), the symmedian point is found.  It will likely 
    be different from the "by eye" dot.  It will certainly be a different fix 
    than that found by following Admiralty advice, particularly with danger 
    > Bill, please answer the following questions for practical navigators.
    > 1)  How are we to know, with certainty, that
    >     [a] the standard deviations of the sights are all the same, and
    >     [b] there is no systematic error
    > Most importantly, the method by which we are to know.   Do you expect 
    navigators to determine the statistical distribution of the observations of 
    each body, so as to know that the standard deviations for each body are all 
    the same?
    > 2) Please justify why using the symmedian point is better than Admiralty 
    advice on the cocked hat.  Or do you only recommend the symmedian point when 
    there is no danger nearby and all of the conditions met?
    > Aside from the pure exercise of applied mathematics, answering these 
    questions should improve the focus and rationale for the symmedian point 
    > Brad
    > View and reply to this message

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