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    Re: No Two-Body Fix Problems
    From: George Huxtable
    Date: 2009 Nov 6, 20:29 -0000

    Peter Hakel wrote-
    
    "It seems to me that by now we have addressed the questions quite
    thoroughly.".
    
    Not for me, yet.  Some matters are still unsettled.
    
    "In general there are two latitudes, one from A+B and the other from A-B.
    Both correspond to genuine solutions."
    Agreed.
    For either latitude, there are two LHA1 (and hence longitude) possibilities,
    hence the four candidates.  This was the original "caveat" that started the
    discussion thread.  Each of the two latitudes has one candidate LHA1 that
    matches the Ho2 (within some round-off), while the other one I called
    "spurious."  That is how we get our two LOP intersections, which I call
    "genuine" solutions.  The spurious ones are "solutions" for the
    GP1-Pole-Ship triangle only, not for the whole problem.
    Agreed.
    
    Peter then addresses the degenerate conditions, which don't concern me here.
    
    He and I agree then, that for each latitude solution (one of which makes
    sense, the other probably doesn't) there are two candidate LHA's, of which
    one must be discarded. They are the two arc-cos solutions of 7.5e, the same
    magnitude of angle but with opposite signs.
    
    He wrote, in [10474] "Once the LHA1 caveat of going from Eq. 7.5e to 7.5f
    was explicitly addressed, John's equations passed that test (and all others
    I did) with flying colors." He and I seem to be agreeing about the existence
    of the same problem, here, though we may be proposing to resolve it in
    different ways. In what way was it explicitly addressed? Was that by the
    cross-check against zenith distance of star2?
    
    But does John Karl agree, too, that something needs to be added, to page 79,
    to enable a real-life navigator to get the right numerical solution to 7.5e,
    to plug into 7.7f and get the right answer? He appears to take a different
    line, in [10464] "...It seems to me that if the computer code simply
    calculates the answers for the two possibles fixes, using A+B and A-B in Eq.
    7.5d, and then lets the navigator pick the closest fix, all possible
    combinations of the locations of the GPs and ship are covered."
    
    Peter Hakel reported in [10414]  "I relay John's comment, that the navigator
    must select C, if the ship is east of GP1, and 360-C, if the ship is west of
    GP1." Yes, something along those lines needs adding to the text, but John
    didn't explain, there, how to discover which side of GP1 the ship actually
    is. After all, where the ship is, is what we're trying to find.
    
    Peter has proposed one way to discard the unwanted longitude, by checking
    against the altitude of the other body. I suggested another, rather complex
    procedure, in [10427]. But the simplest way of all seems to be to check
    whether A-B is less than zero, or A+B is greater than 180 (whichever
    solution was deemed appropriate). This then tells you whether the ship is
    East or West of star1's position, and then the appropriate sign can be
    attached to its LHA.
    
    =================
    
    This leaves aside another matter; whether the cosine formula is indeed the
    best way to determine LHA, when the LHA is a small angle? Wouldn't the sine
    formula be more appropriate, then? That would spoil John's claim that he
    only ever used the one basic formula, however.
    
    George.
    
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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