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    The cocked hat, again. Was: "100 Problems in Celestial Navigation"
    From: George Huxtable
    Date: 2004 Apr 3, 10:23 +0100

    Joel Jacobs wrote-
    >I remember using the least squares method in statistical analysis to
    >establish a line of central tendency, used it to separate the fixed and
    >variable elements of costs in financial analysis, and in a nautical sense,
    >to determine fuel burn at various RPM and speeds. I discovered the bother
    >was NOT worth the effort, and did it visually using a see through or plastic
    >rule which seemed to work equally well.  You visually average the dispersion
    >of the observations around a line, that on the average, would split them
    >equally above and below what be came the line of best fit.
    I think Joel has the right attitude here. The human brain is good at
    instinctive analysis of such data.
    >As far as the cocked hat, did you mention that bisecting the apex of all
    >angles would produce a point that has some mathematical, though not
    >necessarily accurate, MPP? I think that it would work for more than three
    That has been proposed, but applies only to the unlikely situation that
    there's a common systematic error, applying exactly the same to the three
    observations, and no random error, such as would be caused by motion in a
    seaway. It could apply, in smooth conditions, if a mistake was made in
    measuring index error, or if anomalous dip affected every measured
    altitude. In real life at sea, most of the errors will be random ones, and
    that method will not produce the most probable position.
    But I wish to argue here against putting undue emphasis on a calculated
    "most probable position" (MPP) within the cocked hat. There remains much
    misunderstanding, among navigators, about the interpretation of a
    cocked-hat error-triangle. The argument applies just the same to triangles
    formed by three intersecting intercepts, or triangles formed by three
    intersecting compass bearings. Long-standing listmembers will recall that
    this question was given an energetic thrashing a couple of years ago, and
    there's no need to go over those arguments again.
    Making certain reasonable simplifying assumptions, which are-
    1. That all systematic errors have been corrected, so all that remains is a
    random error in the intercept, which is just as likely to be "toward" as it
    is "away".
    2. We can neglect the unlikely event of there being no error at all in the
    Then the likelihood of the true position lying within the triangle is one
    in four, and correspondingly, it's three times more likely to be outside
    the triangle (three times out of four).
    This surprising (to many) conclusion is based entirely on the statistics of
    the problem, and is quite unrelated to the skill of the navigator. A good
    navigator will produce smaller error-triangles than a bad navigator will,
    but still, any such triangle is only 25% likely to embrace the true
    Many navigators still retain the delusion that their true position MUST lie
    somewhere within the cocked hat, and the only question is exactly where,
    within that triangle, it may be. Nothing could be further from the truth.
    That's why a prudent navigator, having drawn a cocked hat on his chart,
    then (mentally, at least) applies a sizeable thumb-smudge to his chart,
    embracing the triangle and a lot more sea besides, and says "I'm somewhere
    here. I think."
    I accept that for some purposes, particularly when navigating by computer
    or electronic chart-plotter, it may be necessary to establish some unique
    mathematical point, about which an error-ellipse is to be centred. In that
    case the MPP, calculated in the way that Herbert Prinz suggests, will
    provide the best answer. What I wish to emphasize is that this MPP should
    not be invested with any magical significance, and the "navigator's
    thumbprint" should be kept firmly in mind.
    In my opinion, for chart-navigation purposes, the intuition of the human
    mind is sufficiently accurate to estimate some sort of most-probable
    position from a cocked hat, without any need for statistical analysis or
    geometry. To that extent, I agree with Joel (if that represents his view).
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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