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    Re: Still on LOP's
    From: George Huxtable
    Date: 2002 May 21, 06:18 +0100

    I've just returned to Nav-L after a couple of weeks "off-watch".
    It's pleasing to note that the cocked-hat affair seems at last to be
    subsiding into an agreed consensus. I have no wish to fan those dying
    embers into life again. But I hope nobody will mind if I make a few
    Trevor Kenchington's library-work is a most valuable contribution.
    Bill Murdoch's method of trimming the problem down to just one dimension,
    to get to grips with its essentials, and Michael Wescott's logical
    responses, show us all how rational discourse between intelligent people
    should be carried on. Also, how valuable a group such as this one can be in
    facilitating it. I hope that Bill is by now quite convinced, but if not, he
    should argue on until he is (one way or another).
    A message from Bill Murdoch, dating back to 6 May and referring to a
    contribution from me, remains unanswered. Because of the time lapse since
    then, I quote it in full-
    >In a message dated 5/4/02 6:24:07 PM Eastern Daylight Time,
    >george@HUXTABLE.U-NET.COM writes:
    >> Bill Noyce made a perceptive contribution a few days ago, about systematic
    >> errors in celestial observations that can increase the probability of the
    >> true position lying within the cocked hat. This happens because those
    >> errors expand the cocked hats to surround the true position.
    >I am not sure that is correct.  A three body fix using celestial navigation
    >puts three small circles on the surface of the earth.  If all were perfect,
    >the three would intercept at one point.  Because all is not perfect they
    >intercept making a triangle.  Draw three circles intercepting to form a
    >triangle in the middle.  If you are like me, you drew three circles of about
    >the same size intercepting in a total of six places with a small triangle in
    >the middle with all three sides bulging outward.  You pointed at the small
    >triangle in the middle, the one with three convex sides, and said, "I am
    >here."  That is true if all three LOPs are away from their bodies.
    >Surrounding that triangle are three more triangles with two convex sides and
    >one convex side.  You would be in one of them if two of the LOPs were away
    >and the third was toward.  To see the other two cases, you have to redraw the
    >circles moving the centers around to leave triangles with either three
    >concave sides or two concave sides and one convex side.
    >If the usual systematic error is to tilt the sextant and record an
    >erroneously large reading, the circles will be too large.  In some cases that
    >will make the triangle larger.  In other cases it will make the triangle
    >Bill Murdoch
    I haven't quite grasped the details of the geometry Bill is describing, and
    ask him to clarify it further, if he is still asking for further comment
    from me. It may have since been overtaken by Bill's growing enlightenment.
    However, it gives me a chance to emphasise something I left rather unclear
    in earlier mailings, which is this:
    Changes in a cocked hat due to systematic errors in measurement (such as
    sextant-tilt in astro LOPs, or variation error in landmark
    compass-bearings) differ greatly depending on the way those lines are
    scattered around the observer's horizon.
    The simplest example is that of variation affecting a set of three
    compass-bearings, all in the same sense. Then, as I said in an earlier
    mailing, if that error exceeds the instrumental scatter, and if the
    landmarks are spaced 120 degrees apart, the apparent bearings are all
    shifted in such a way as to enlarge the resulting triangle, and ensure that
    it must embrace the true position.
    However, it's unusual to be in a position where there are landmarks
    surrounding a vessel in that way. It's more common to be running along a
    coast, taking compass bearings of three landmarks which are much more
    closely spaced, for example 60 degrees apart. If a variation error causes
    the bearing lines from those landmarks to be all displaced in the same
    sense, and by more than the instrumental scatter, then every cocked hat is
    now guaranteed NOT to embrace the true position.
    I think (but haven't attempted to prove) that the distinction between these
    cases depends on whether the observer's true position is either within or
    without the triangle which joins the three landmarks.
    Similar situations occur with LOP's from sights of three celestial bodies.
    My thanks to Clive Sutherland for emphasising this matter to me, off-list.
    I am unsure whether this is relevant to the point that Bill Murdoch was making.
    George Huxtable.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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