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    Re: Making an impact with hav-Doniol (collection of prior posts)
    From: Hanno Ix
    Date: 2015 Jun 22, 04:41 -0700

    Gary, Ken

    the hav - Doniol story is even older yet:

    1.
    navlist.net/m2.aspx/g27982

    In June 2014, Greg found an existing version of Doniol, one that one could call sine-Doniol.
    As he noted in this posting :

    R. Daniol (1955)  formula :

    Sin h = n - (n+m)a

    n = Cos(L-d)

    m = Cos(L+d)

    a = haversine LHA

    It still required at least two tables, if not three, but still had sign complications in the trig functions. However it was mathematically strictly applicable to all permutations of L, d and LHA and needed only one multiplication. So it was a very welcome simplification of the Hc formula for sight reduction. Greg went through several examples that demonstrated these advantages very clearly.


    In Nov 6, 2014 I re-wrote the sine-Doniol formula into a haversine - only version and compared it with the sine - Doniol in  the first of these postings addressed to Greg:

    hav(s2s) = hav(L-d) - [ hav(L-d) + hav(L+d) - 1 ]* hav(LHA).

    where I used s2s for ZD for my own convenience, and I included an example. This, then, is the oldest hav - Doniol version. Greg and I were very happy about this discovery.

    Greg is very practical: he prefers to proceed in well defined steps as his hav - Doniol examples in  this posting show. With these examples he compacted hav - Doniol into a form similar to the step-by-step procedure of the original sine - Doniol:

    n = hav (L- d)
    p = hav (L+d)
    q = n + p
    hav(ZD)  = n + (1 - q) * hav( LHA )

    which has become the preferred version. It easily can be executed on an index card as Greg likes to do on his RIC's = "Rudzinski Index Cards".

    4.

    My 2-page haversine table was introduced several days later:navlist.net/m2.aspx/g29172
    and the azimuth diagram almost 2 years before.navlist.net/m2.aspx/g20982
    Some other proposals I made in this context, for instance the Vedic multiplication scheme, did certainly not find approval at all.

    5.

    Comment: there was no particular mathematical ingenuity on my part involved - just certain insights into the needs of the navigator which are mainly issues of dis-ambiguity and simplitiy. The decisive steps were 1. and 2. above and perhaps my azimuth diagram. Since we discovered hav - Doniol Greg spread the word in a tireless effort.

    Strangely enough, there were authors ( e,g, Willis ) who developed a very similar formula but did not recognize the advantages.


    H



       
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