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    Re: Position lines, crossing.
    From: George Huxtable
    Date: 2006 Dec 11, 15:16 -0000

    Bill, in [NavList 1864], quoted from Dutton's, about intersecting
    position lines. I have a healthy respect for Dutton's.
    
    That passage (para.3018 in his 15th edition, 3009 in my 12th edition)
    is dealing with the situation of a systematic error, the "constant
    errors" that are referred to, with negligible random scatter . And if
    you had a way of knowing that was the situation, you could apply that
    recommended procedure. But from a single 3-body fix, how could you
    possible tell whether any discrepancies stemmed from "constant error",
    or from random scatter? They are entangled together, in a way that's
    almost impossible to separate.
    
    Our analysis, putting only 1 in 4 inside the triangle, assumed
    entirely random scatter, and no "constant error", and specifically
    said so. It's the very opposite to what that section from Dutton's was
    considering, as quoted by Bill.
    
    In his analysis of the problem that Guy Schwarz presented, in [NavList
    1848], Geoffrey Kolbe was able to deduce , with reasonable conviction,
    that all six observations may have been subjected to just such a
    common error, because shifting them all "away" by the same amount gave
    a much more consistent picture of their crossings. That may indeed
    have been a valid deduction from that evidence of six altitudes. But
    to make it with any confidence, you need a sufficient number of
    observation, to over-determine the answer and examine any conflicts
    between them. That's the valid point that Frank has made. You just
    can't do it, on the basis of a single cocked hat.
    
    Dutton's is not infallible about these matters, though. In para 2016,
    in my 12th ed., referring to position lines, he writes "In actual
    practice they will seldom intersect at a point, but will produce a
    small polygon, often referred to as a "cocked hat", which usually
    contains the position of the ship." On the contrary, we have seen that
    where there are no systematic errors, usually, (three times out of
    four) a triangle will not contain the position of the ship. On that,
    Dutton's is quite wrong.
    
    =====================
    
    There may be a bit of a question about Guy's analysis, seeing that
    John Cole claims differing answers, which might possibly explain the
    systematic discrepancies that have been seen.  It would be useful for
    someone to set down for us the original problem, as set.
    
    Nevertheless, Guy's plot of position lines, in [NavList 1848] with its
    six crossing lines, can be instructive, to be pondered on by anyone
    still arguing that a cocked hat must contain the ship's position. That
    plot shows a jumble of 6 lines, from which 20 possible cocked-hats can
    be constructed, from each possible set-of-three, if the other three
    are ignored. Consider a few of those possiblilities. Imagine that
    none of Kochab-Rasalhague-Venus been observed, and the position had
    been deduced from Altair-Alpheratz-Moon only. Those three happen to
    coincide nearly in a single point (just like one of Henry Halboth's,
    perhaps) so there we have a tiny cocked hat, establishing the vessel's
    precision precisely.
    
    Compare that with the result if, instead, Kochab-Rasalhague-Venus were
    adopted, and the other three ignored?  A quite different triangle
    results, far, far bigger. It happens, in that case, to enclose our
    earlier triangle; but sometimes, two such triangles will have no area
    in common whatsoever, and then, it's quite impossible for the same
    vessel to be in both triangles. Such as Kochab-Alpheratz-Moon, and
    Rasalhague-Altair-Venus, for example.
    
    The divergences, between those triangles, nicely illustrate another
    point that Frank Reed made.
    
    =====================
    
    Robert Eno asked-
    
    "So here is my question: for purposes of practical navigation at sea,
    does
    one really have to take into account the statistical probabilities of
    the
    actual location of the fix,...or is it
    sufficient enough to do what countless navigators have done for
    generations:
    take the fix from the centre of the cocked hat and/or the centre of
    the
    "smudge"?
    
    I believe that the latter it is good enough for practical purposes."
    
    Yes, of course that's what you have to do. There really isn't an
    alternative. And in assessing the size of that smudge, all I ask is
    that common sense is used, some experience of how good one's
    observations are likely to be in the current conditions, and not blind
    adherence to the outlines of a particular cocked hat as the limit of
    error. I think there's no disagreement on the matter, between Robert
    and me.
    
    And really, there's no call, and seldom an opportunity, to make any
    "statistical analysis".  Never. Just a bit of common sense is all
    that's required. "With eyeball and thumbprint" (perhaps a good title
    for a navigator's biography).
    
    Robert added-
    
    "In terms of nearshore navigation where one is likely to encounter
    dangerous
    reefs, astro navigation is the last thing I would want to employ.
    Shore
    bearings would be a more accurate means of keeping one's distance from
    dangerous ground and better yet, radar."
    
    Well, that's a point of view you can take today. But as little as 20
    years ago, mariners were having to approach reef-strewn coasts in
    parts of the world which were badly lit and badly marked, and had no
    other option. In thick weather, they were blind.
    
    George.
    
    contact George Huxtable at george@huxtable.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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