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    Cocked hats, again.
    From: George Huxtable
    Date: 2007 Mar 13, 21:30 -0000

    I hardly ever find anything to disagree with in Gary LaPook's
    contributions. They are usually full of sense.
    
    That applies, too to most of his 6th March posting, Navlist 2236,
    labelled "resolution of systematic error". Though actually, what he
    discusses there is the situation of entirely random error, where all
    systematic error has been corrected out.
    
    But Gary concludes with this-
    
    "Again, no one
    is suggesting that the position of the observer is at the center of
    the triangle but this represents the center of possible positions of
    the observer.  In fact, the position of the observer will be outside
    of the triangle often but I don't agree with the three out of four
    allegation. Counter intuitively, the smaller the triangle the more
    likely that the position of the observer is outside the triangle! If
    you think about it, this should be obvious. Using   reducio ad
    absurdum, think about a triangle only one inch in size, it would be
    impossible for the observer to be within the triangle. At the other
    extreme, a very large triangle with all of the displacements of the
    LOPs from the center of the triangle equal to 3.3 NM (3.3 sigma's,
    linear sigma's are slightly different than circular sigma's, see
    Bowditch), the only place that the position of the observer could be
    is at the fix in the center of the triangle!"
    
    Here, Gary is wrong. The statement, that "with entirely random errors,
    three times out of four the triangle will not embrace the position of
    the observer", is precisely correct, and based on irrefutable
    statistical arguments. If Gary doesn't accept this, unlikely though it
    may seem to him at first sight, he should scan through previous
    discussion of cocked hats on this list, as Geoffrey Kolbe suggested.
    If he remains unconvinced, we can go through the arguments once again,
    until one way or another, either he accepts it, or else he convinces
    us otherwise.
    
    The best way to consider the matter is not to take a particular
    triangle, and then consider "where can the true position be?", but to
    take a true, known position of an observer, and three celestial
    bodies. Now plot in his vicinity a set of three position lines that
    have been displaced by Gaussian amounts, toward or away from the GPs
    of the bodies in question. (Indeed, the argument doesn't rely on a
    Gaussian distribution, just equal numbers toward and away). And then,
    with the same observer position, and the same three bodies, plot
    another triangle, with different displacements, and another, until you
    are tired of it. The resulting triangles will be different each time;
    some will be large, some will be tiny, some long and thin, others
    nearly equilateral. And in the end, if you check enough triangles, you
    find that 25% of those triangles will embrace the true position that
    you started with.
    
    It doesn't matter how skilled or unskilled the observer is; it remains
    true. For a skilled navigator, of course, the triangles will indeed be
    smaller on average, which is where his skill shows itself, but still,
    only 25% of those smaller triangles will include the true position.
    
    George.
    
    contact George Huxtable at george---.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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