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    Haversine formulae for Great Circles
    From: Lu Abel
    Date: 2001 Nov 17, 12:08 PM

    Almost every celestial text provides the formulae for calculating great
    circle distance and direction between two points.  This is not surprising
    since the trigonometry is identical to that of sight reduction.  If one
    takes any standard celestial reduction method and substitutes the starting
    point for the AP and the ending point for the body GP, Zn gives the
    direction and 60*(90-Hc) gives the distance.
    
    A couple of years ago this august group was of great help in answering for
    me an obvious question I've never seen answered in any standard navigation
    text, namely how to calculate the L/Lo of intermediate points.
    
    Recently I've done a bit of further digging on great circles.  I learned
    that while the law of cosines formula typically used for Hc calculation is
    trigonometrically correct, it can produce incorrect answers on a computer
    or calculator when the starting and ending points are close because
    computers and calculators express numbers with a limited number of
    significant digits.  An alternate is the "Haversine" formula, called by
    that name because of its haversine-like terms  [recall that hav(x) =
    (1-cos(x))/2 = sin^2(x/2) where "^2" means squared ]
    
    The haversine distance formula goes as follows:
    
    L1, Lo1 are L/Lo of the starting point, L2/Lo2 are the L/Lo of the destination
    
    DLat = L2 - L1
    DLo = Lo2 - Lo1
    
    A = sin^2(DLat/2) + cos(L2) * cos(L1) * sin^2(DLo/2)
    
    Distance  =  120 * arcsin (sqrt (A))
    
    
    For a great writeup on the haversine distance formula, see
    http://mathforum.org/dr.math/problems/neff.04.21.99.html (note no "www" at
    beginning)
    
    Having learned about this formula, I'm going to guess it's used in the
    majority of GPS sets, since they're often calculating small distances (like
    distance-to-go when approaching a waypoint)
    
    The standard reference on calculating great circles via the haversine
    formula seems to be R.W. Sinnott, "Virtues of the Haversine", Sky and
    Telescope, vol. 68, no. 2, 1984, p. 159.  I've seen it mentioned in several
    writeups of the haversine distance formula.
    
    I can't access a copy to see if Sinnott provided formulae beyond the
    distance formula, so here's my question for this group:
    
    Are there equivalent "haversine" formulae for initial direction and the
    L/Lo of intermediate points?
    
    Thanks
    
    Lu Abel
    

       
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