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    Re: Position lines, crossing
    From: Geoffrey Kolbe
    Date: 2006 Dec 10, 13:20 +0000

    George Huxtable asked [NavList 1853]
    "I hope that Geoffrey isn't
    going to claim that it's bounded by his new, much-reduced, triangle."
    No George, that is why my analysis and statement of the final deduced
    position finished with the word "- roughly", in the sure and certain
    knowledge that the statistics of how good that position is were about
    to be thrashed out :-)
    George is quite correct that the probability of the position being
    inside the cocked hat of a three position fix is just 0.25, assuming
    systematic errors are small compared to random errors. That is a
    carved-in-stone fact which, as George laments, many people have a
    hard time grasping.
    Peter Fogg wrote [NavList 1856]
    "Geoffrey's solution has the appeal of
    simplicity, although there seems to be little point in doubling the
    sight reduction needed to produce 6 LOPs only to reduce them back to 3
    in this rather arbitrary fashion."
    Perhaps Peter has not appreciated that the purpose of amalgamating
    sights of bodies in opposition is to remove systematic error. This is
    a time honoured method, both in surveying and in navigation and there
    is nothing "arbitrary" about it.
    The justification for assuming that the separation of the
    Moon-Rasalhague pair of sightings (which were almost exactly in
    opposition) was due to a systematic error of some sort is the fact
    that the final "cocked hat" ended up so small after applying the same
    systematic error to all the sightings (which is effectively what I did).
    Have a look at "Celestial Navigation for Yachtsmen" by Mary Blewitt.
    (I love that title - the assumption being that yachtswomen do not
    need a textbook in navigation :-)) She demonstrates exactly the
    methods I used to reduce a jumble of confusing sights down to
    something manageable and tractable.
    "Hints for Travellers", published by the Royal Geographical Society,
    (about which there was some discussion recently when Frank Reed
    pointed out that the 1906 edition was up for auction on ebay and I
    successfully bid on it), details the methods by which a surveyor may
    fix his (or her) position to within a few seconds of arc. The
    underlaying principle is to take sightings of bodies in opposition
    (North-South for latitude, East-West for longitude) in order to
    reduce systematic error - principally in how level the theodolite is.
    I recently chatted with a retired Royal Air Force navigator who
    served on Vulcan bombers. He explained to me that their method was to
    take sightings on stars immediately ahead and aft of the aircraft to
    fix distance travelled and then to take sightings on stars port and
    starboard to fix direction.
    When precision is required, it is common practice to take sightings
    of bodies in opposition, then amalgamate them to eliminate systematic error.
    Geoffrey Kolbe
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