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    Re: Azimuth and Declination formulae
    From: Lu Abel
    Date: 2005 Jul 19, 21:05 -0700

    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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