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    Re: Azimuth and Declination formulae
    From: Henry Halboth
    Date: 2005 Jul 20, 11:55 -0400

    Many thanks Lu - you saved me the trouble. There is probably no modern
    use to the formul;a posted - just, in may opinion, an interesting
    historical note and perhaps part of one's math education.
    
    On Tue, 19 Jul 2005 21:05:42 -0700 Lu Abel  writes:
    > Whoa on the haversines.  It's not half of a sine, it's half of a
    > versine.
    >
    > A versine (x) = 1 - cos (x).   Note that vers (x) has a range from 0
    > to 2.
    >
    > Haversine (x) = vers (x) / 2.   This just makes hav (x) have a range
    > from 0 to 1.
    >
    > The whole reason for versines and haversines was to allow sight
    > reductions to be done using logarithms (and therefore the requisite
    > multiplications become additions); but logs are not defined for
    > negative
    > numbers, hence the need to shift everything to have a positive
    > value.
    >
    > Versines and haversines can also be expressed in terms of sine
    > squared,
    > vers (x) = 2 sin^^2 (x/2).
    >
    > As a side note, the traditional formula for the great circle
    > distance
    > between two points breaks down into finding the difference between
    > two
    > nearly equal large quantities for small distances.  This can produce
    > inaccurate answers because calculators and computers only carry out
    > calculations with a limited number of digits.  The equivalent
    > haversine
    > formula is well behaved, subtracting two small numbers.  Therefore
    > all
    > GPS's actually use the haversine formula for calculating the
    > distance
    > between two points.
    >
    > Lu Abel
    >
    > Peter Fogg wrote:
    > >>From: Henry C. Halboth
    > >>A bit more complicated, but generally employed with the Time Sight
    > is ...
    > >>
    > >>hav Z = sec ho x sec L x sin 1/2S - ho x sin 1/2S -L, where ...
    > >>
    > >>Z = azimith, named according to Latitude + meridian angle, E or W
    > >>ho = corrected altitude
    > >>L = Latitude
    > >>pd = polar distance
    > >>S = ho + L + pd
    > >
    > >
    > > Interesting. Presumably 'hav' stands for haversine, which I
    > vaguely recall
    > > is a half sine? And 'sec' is secant? I don't know what that is.
    > >
    > > What do I need to be able to use this formula? Scientific
    > calculators I
    > > have. Do I need tables of havesines and secants?
    > >
    > > What is the advantage of this formula?
    > >
    > >
    >
    
    
    

       
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