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    Re: Cocked hats, again.
    From: George Huxtable
    Date: 2007 Mar 15, 15:24 -0000

    
    
    
    Gary LaPook wrote, about the probability of the true position of an
    observer lying within a "cocked hat".
    
    | Well George, I always was partial to "T's" and "A's."
    |
    | I follow you argument and can close my eyes and visualize what you
    are
    | saying.
    |
    | How's this for another description of your point.
    |
    | Draw three LOPs surrounding the position of the observer. If there
    had
    | been no random errors, the three LOPs would have plotted at a point
    at
    | the location of the observer. Now indicate on the LOPs with an arrow
    | the azimuth of the body, lets say all pointing outward so this
    | represents the "TTT" case. Now if you flip one of the LOPs and make
    it
    | an "A" it will now plot closer to the interesection of the two
    | remaining "T" LOPs forming a smaller triangle over in that corner of
    | the original "TTT" triangle and the position of the observer wil not
    | be within the new small triangle. Then you can conceptionalize this
    to
    | the other possible combinations and come up with the 1 in 4 ratio.
    |
    | Do I understand you point?
    
    =====================
    
    From George.
    
    Yes, we agree. Can we take it, then, that Gary now accepts that 1 in 4
    figure?. In that case, the matter's settled.
    
    Except that a minor point remains. If the three azimuths can be
    enclosed within a 180-degree arc (and we make the middle-radius of
    that fan of arcs corresponds to the middle letter of our three-letter
    code) then the AAA and TTT conditions will not enclose the true
    position. Instead, the ATA and TAT conditions will do so. Still 1 out
    of 4, just the same.
    
    Separating those two scenarios, of azimuths that embrace more than 180
    degrees and those that embrace less, is the dividing case of azimuths
    that are just 180 degrees apart. As Fred points out, in that case the
    triangle degenerates to a straight line. It isn't a triangle at all,
    and there's little point in considering that special case in any
    detail, though I'm a bit puzzled how Fred manages to make 7
    possibilities out of it.
    
    Robert Eno, in Navlist 2343, points to the relevance of other aspects
    of Gary's posting, and I agree with all he says. Of course, when you
    have to plot a point to represent your position, the centre of the
    triangle is as good as any. What's important, though, is that the
    navigator should have some notion of the uncertainties involved when
    he plots that point.
    
    In Navlist 2330, I picked on Gary's disagreement with the 1 in 4
    statistics, because that was all I found to question. I think that the
    1 in 4 argument is worth getting right, because it's conclusions are
    so simple, and it conflicts so much with what many navigators assume,
    or have been taught, and because the consequences of misundertanding
    the matter could be so serious.
    
    Fred contributed- "A small triangle suggests that sigma is small for
    each line, and a large
    triangle that sigma is large". I agree that a large triangle implies
    that there's a lot of scatter, but the contrary isn't true of a small
    triangle; it depends a bit on what Fred means by "suggests".
    
    Indeed, a single cocked-hat gives little clue to the possible scatter
    in the true position. Just by chance, it might well happen to be a
    tiny triangle, but that doesn't allow a navigator to deduce that the
    resulting position is a precise one. It's only after assessing the
    sizes of a large number of such cocked hats that a navigator could
    come to such a conclusion. Not a practical proposition really; in its
    place he has to fall back on common sense and experience, in
    estimating what his observation errors are likely to amount to.
    
    Robert also raises the question of how this matter is presently taught
    in navigation classes, and I would like to take that a bit further,
    and ask what Navlist members have been taught, and what they believed
    to be the truth, before this matter ever came up for discussion on
    this list (or its predecessor).
    
    When I was taught navigation in an evening class, over 40 years ago,
    the matter was simple. The triangle, whether between compass bearings
    of landmarks or between astronomical position lines, gave you your
    position. You must be somewhere within it, if you didn't know exactly
    where. To be absolutely safe (we were told), if there was a nearby
    danger, you had to assume you were in that bit of the triangle that
    lay closest to the danger. At the time, I accepted that nonsense. Not
    that I was less argumentative, or more docile, in those days, but I
    knew no better.
    
    What enlightened me was an Open University programme on TV, not on
    navigation, but on statistics, in which the triangle between magnetic
    bearings was used as an example of statistical methods. The
    one-in-four conclusion was a real eye-opener.
    
    Robert asks, of navigation teachers- "Do you teach them to take the
    fix at the intersection of the LOPs and/or centre of the cocked hat or
    do you hose them down with cold statistical sea water just as they are
    beginning to grasp the fundamentals?". That's a somewhat tendentious
    way of putting it. It's not asking a lot, of teachers or students, to
    put across the simple message as follows. "Plot a point, at the centre
    of the triangle, but be aware that the true position could lie well
    outside that triangle."
    
    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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