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    Re: No Two-Body Fix Problems
    From: Peter Hakel
    Date: 2009 Nov 6, 00:15 -0800
    John, please see below.


    From: John Karl <jhkarl---.net>
    To: NavList <navlist@fer3.com>
    Sent: Thu, November 5, 2009 6:51:52 PM
    Subject: [NavList 10469] Re: No Two-Body Fix Problems


    Peter,  OK, good.

    I should also mention that there's no need to use the other solutions
    to the inverse cosines in Eq. 7.5a&b (i.e., like 180-A) because those
    solutions just switch (A+B) and (A-B) when substituted into cos(A-B)
    in Eq. 7.5d (maybe this isn't obvious).  But that doesn't matter since
    we're computing both solutions anyhow.

    ===========================

    PH:
    This detail was in fact quite clear.



    ===========================

    And I agree that the equations should be perfectly well behaved in all
    practical cases.  They only blow up at 90d declinations, 90d
    altitudes, or zero D12 -- cases that will never occur in practice.
    Also the equations will give the correct result for osculating LOPs.
    In that case B=0, and the two solution for (A+B) and (A-B) will give
    the same answer.

    So these equations should be straight forward to implement in computer
    code.

    ===========================

    PH:
    After a few iterations I arrived at an Excel spreadsheet showing that your equations are much more robust than you think.
    D12=0 defines an ill-posed problem, so that is not a fair test.  However, even for declinations and altitudes of 90 degrees, osculating
    LOPs (i.e. B=0 or 180), and below horizon sights (Ho<0), your procedure has passed all tests that I put it through so far.


    Peter Hakel



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