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    Re: Position lines, crossing.
    From: George Huxtable
    Date: 2006 Dec 10, 10:25 -0000

    List members will appreciate the difficulties we are faced with in
    trying to convince others about the realities of a cocked hat.
    Already, three have put their heads up in defence of their cherished
    illusions; Peter Fogg, Bill, and Robert Eno. No doubt, others will
    join them. We've seen it before.
    
    We need to be clear about what we are telling these people, that they
    are so reluctant to accept. But first, let be clear about what we are
    NOT saying.
    
    Nobody is decrying the taking of a round of three sights, to get a
    resulting triangle, in terms of three sextant altitudes or three
    magnetic bearings. It's the staple diet of the navigator. There's
    nothing wrong with plotting your derived position on the chart, at the
    "centre" of the triangle, or even at one edge or corner if there's a
    local danger that you need to clear. You have to plot it somewhere,
    and those positions are as useful as any.
    
    The problem comes in understanding the result, and the meaningfulness,
    of what you have just done.
    
    If you think that your real position is certain, or even likely, to be
    contained within that triangle, you are wrong. It isn't. If you think
    that you can be no nearer to the danger than the nearest corner or
    edge of the triangle, you are wrong. If you think that a small
    triangle indicates for sure that you have deduced a precise position,
    you are wrong. Those delusions may be simple to put into a textbook
    and teach to simple seamen; the authors of those textbooks may or may
    not have understood the realities of the matter. But they have been
    purveying dangerous nonsense. Those delusions are widely held, and
    bravely defended by their adherents, but that does not make them
    right.
    
    ===========================
    
    Robert Eno suggests "that he would simply assume he is in the centre
    of the cocked hat
    | and/or he would draw a big circle or ellipse or whatever, around the
    | profusion of criss-crossing lines and aim for the centre of mass;
    then he
    | would carry on as normal."
    
    Well, he makes two proposals there, that are quite contradictory. If
    he "assumes he is in the centre of the cocked hat", then what's the
    purpose of the "big circle or ellipse"? And I suspect that Robert is
    really advocating something a bit different. That a navigator plots a
    point on the chart at the centre of the cocked hat (where else, after
    all?) but without any assumption that he is exactly there; then draws
    a smudge around it, either on the chart or at least in his head, and
    accepts that he will be somewhere within that. If that represents
    Robert's view, then we agree.
    
    ==========================
    
    Peter Fogg wrote-
    
    "As to George's insistence that the ACTUAL position is not necessarily
    located at the centre of a triangle, or other intersecting LOPs, this
    does not seem especially helpful... Will this knowledge enable the
    calculation of a more
    accurate position?"
    
    No, certainly not. But that isn't the point. The point is, to dispel
    the dangerous illusion that by creating a cocked hat you have
    discovered a precise position, or set secure boundaries about where it
    must be.
    
    "The value of the nominated position at the centre of position lines
    is
    that it is the only possible CALCULABLE position."
    
    Indeed. I do not criticise the use of it. What's the alternative?
    
    "Saying that the
    actual position could be somewhere else doesn't change this, or assist
    the goal of position finding in any way that I can see."
    
    Peter misses the point. If alternatively, he assumes that his position
    must be contained within such a triangle, that is WRONG, and
    dangerously so. My aim is to dispel that illusion.
    
    "Why do we want to define a position, anyway? Usually it is not to
    know
    where we are, since after sight reduction and plotting in any sort of
    moving craft we are no longer back there any more. Usually the
    calculated position is then used to run forward our track since then,
    a process largely beyond precision."
    
    Yes, there's a bit of extra error to add, where positions need to be
    "run on" with some DR. But why should that affect the proper
    assessment of a 3-object fix?
    
    "Moving this calculated position closer to identified danger seems
    like
    simple common sense."
    
    Maybe, but assuming that the vessel can't be closer to that danger
    than the bounds of the triangle is sheer folly.
    
    ========================
    
    And Bill wrote-
    
    "No
    systematic errors or personal bias, you are *probably* in the cocked
    hat.
    No one-in-four statistics as related in my texts.  With systematic
    errors,
    then you may well be outside the cocked hat according to my texts."
    
    Here, Bill is simply wrong. With no systematic errors, you are
    probably outside the cocked hat; three times as likely, indeed, as
    being inside it. It isn't the systematic errors that put you outside
    the triangle, in general; it's the random scatter. Bill's texts may
    not refer to this matter, in common with most others, but that doesn't
    mean that it's wrong.
    
    "If
    I have done my homework before a long voyage and do not find personal
    or
    systematic problems, or instrument error, then I might tend to believe
    that
    after a month at sea that would remain true and trust a position
    within the
    cocked hat within certain limitations."
    
    That would be a deluded belief; but it depends on the "certain
    limitations". Again, it's not the systematic errors that put you
    outside the cocked hat, in general. It can be quite the opposite. Let
    me explain that point.
    
    Consider a simple example, of compass bearings taken from a stationary
    vessel of three distant lighthouses, at azimuths 0 deg, 120, and 240.
    Let's say that the sea-state gives rise to a scatter in the observed
    bearings, of up to, say, 3 degrees either side. Clockwise and
    anticlockwise errors are equally likely. With no systematic error,
    repeated rounds of 3 bearings will give rise to a number of differing
    triangles, of which one quarter will embrace the true position. These
    are the occasions when, by chance, all three bearings have errors in
    the same sense: either all clockwise, or all anticlockwise. It may be
    helpful to sketch out some possible triangles, to visualise it.
    
    But now, introduce a big systematic error. Let's say that in error you
    applied variation, which was 5 degrees West, the wrong way, adding it
    to the observed bearing rather than subtracting. Now, every
    "corrected" bearing is in error, too great by 10 degrees, shifted
    clockwise. Every time you plot the corresponding reciprocal bearing
    from each lighthouse, that too will be shifted clockwise by 10
    degrees. That will overwhelm the scatter, in such a way that now,
    every such bearing will aim off from the true position of the vessel,
    always the same way (clockwise), by 10 degrees on average, but varying
    between, say, 7 and 13. That will considerably increase the size of
    our cocked hats. But more relevant, because those enhanced errors are
    now all in the same direction, every one of those cocked hats now
    embraces the true position of the vessel! Sketch it out, to see it.
    
    So this is an example where introducing a systematic error actually
    increases (and considerably so) the chance of a cocked hat enclosing
    the true position. It happens, because the size of each cocked hat has
    been so greatly expanded. A similar effect can occur when plotting
    celestial positions.
    
    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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