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    Re: Direct methods
    From: George Huxtable
    Date: 2007 Nov 4, 11:55 -0000

    Responding to my comment in 3760-
    | > Equations 7.5a to 7.5d take you to the latitude
    | > of the crossing; indeed, the two latitudes, that you have to choose
    | > for the two crossings. No analysis will choose for you which of those
    | > you are at
    Ronald van Riet replied-
    | For starters, if you don't have a rough idea where you are, you are a
    | very poor navigator, so taking your dead reckoning position wil be
    | sufficient for eliminating ambiguity.
    Yes, that's exactly why I added the words (which Ronald omitted) "..., and
    it's hardly ever a problem to do so."
    | If you want a formal way of doing so, shoot a third star.
    | IMHO there is no mathematical way to take away the ambiguity with just
    | the two star observations, just as it is impossible to define a plane
    | by just two points.
    Yes, of course, that's the case. Two position circles, if they intersect at
    all, intersect twice. There is no analysis of the numbers that will decide
    between the two solutions. Which was exactly what I wrote, above. That
    ambiguity is unavoidable. It can only be resolved by the navigator using
    commonsense, knowing roughly where he is. I doubt if Ronald and I disagree
    at all about any of that.
    The point I was trying to make, and which I suspect Ronald might have
    missed, is this. The equations (7.5e, 7.5f) that have been provided in the
    book to determine longitude from cos LHA1, add an EXTRA level of ambiguity
    to the unavoidable one that Ronald and I have discussed above. For EACH of
    the two unavoidable ambiguous latitudes those equations produce TWO possible
    longitudes. And it isn't always easy to determine which of those longitudes
    to choose, by commonsense. It gets particularly difficult if the azimuth of
    a star is anywhere near North or South of the observer. But THAT ambiguity
    IS possible to resolve, and without needing to refer to any DR position. The
    procedure described in my earlier attachment on intersecting circles is one
    way to do it, but it's not very elegant, and I would be interested to learn
    of a better one.
    How it works is this- First take one of the two possible values for latitude
    that have been determined, say for the northernmost of the two
    intersections. Draw (in your mind) that line of latitude. We know that it
    intersects each position circle twice, and we can calculate the four
    longitudes of those intersection points, using an equation such as 7.5f,
    applied to EACH circle. Remember that when 7.5f calculates LHA1 from its
    cosine, that has two solutions, equal positive and negative angles. But one
    of those longitudes, from circle 1, will be exactly equal to one of those
    longitudes, from circle 2. That must be the longitude of the wanted common
    intersection, between the two circles and the line of latitude. The other
    longitudes are discarded.
    Next, take the other possible answer for latitude, and calculate the
    longitude that corresponds to that, in just the same way.
    I hope that Ronald will re-read my posting and perhaps think again.
    contact George Huxtable at george---.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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