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    Re: Direct methods
    From: George Huxtable
    Date: 2007 Nov 4, 21:43 -0000

    Michael Dorl wrote-
    
    | Back in 1999, I sat down and worked out the spherical trig for this
    | problem.  It may be found at
    |
    | http://ns.doit.wisc.edu/~mdorl/Sight%20Reduction%20write%20up.doc
    |
    | I am pretty sure it handles all the quadrant problems  I did not consider
    | problems at angles when the trig functions misbehave.
    
    ====================
    
    That's interesting, and timely. Except in detail, it's on exactly the same
    lines as the procedure quoted in John Karl's book. Understand one , and you
    will understand the other.
    
    However, one of those details is a silly error in Michael Dorl's program, in
    which his final equation for Cos(T), on the last page, should  not be-
    
    cos(T) = [Sin(AltJ)-Cos(DeclJ)Sin(Lat)] / Cos(DeclJ)Cos(Lat)
    but instead-
    cos(T) = [Sin(AltJ)-Sin(DeclJ)Sin(Lat)] / Cos(DeclJ)Cos(Lat)
    
    Mike will kick himself when he sees it! Making that change, and allowing for
    different notation, his equations for Sin(Lat) and for Cos(T) then
    correspond EXACTLY with John Karl's 7.5d and 7.5e.
    
    In both cases, there are two solutions, for lat and cos(T) depending on
    whether an Angle, (or B) is to be added to or subtracted from a Bearing (or
    A), and both those solutions should be calculated, to provide two lats, each
    with an associated cos(T). Both treatments are identical, up to that point.
    
    But the other detail is an important one. Right at the end, Mike then goes
    on to state a rule to define whether the value for the angle T,
    corresponding to the LHA of the Westmost of the two stars from the observer,
    should be taken as the positive or the negative result of Cos(T). It's
    really judging whether the vessel happens to be West or East of the meridian
    of that star. Mike does this with a reasonably simple test of two angles
    that have already been calculated, and I think he gets it right. That then
    produces, for each of the two latitudes, an appropriate longitude. That's
    the step that's missing from John Karl's analysis, as a result of which, he
    leaves an EXTRA ambiguity in the longitude, as we've been discussing,
    resulting in two possible longitudes for each latitude.
    
    George
    
    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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