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    Re: Spherical Law of Cosines
    From: Dave Weilacher
    Date: 2002 Oct 18, 10:12 -0700

    Thanks.  Very pleased with you.
    
    Dave
    
    On Fri, 18 Oct 2002 10:05:29 -0700 Dan Allen  wrote:
    
    > On Friday, October 18, 2002, at 07:53 AM, David
    > Weilacher wrote:
    >
    > > Please define little (a), (b), (c), and (ab),
    > for me.
    >
    > With pleasure!
    >
    > cos(c) = cos(a) * cos(b) + sin(a) * sin(b) *
    > cos(ab)
    >
    > a is the length of the first side of the
    > spherical triangle.
    > b is the length of the second side of the
    > spherical triangle.
    > ab is the angle between the two sides a and b.
    > c is the length of the side opposite the angle
    > ab.
    >
    > Here is an example of a great circle distance
    > between
    > San Francisco (SF) and Salt Lake City (SLC).
    > All angles
    > and lengths are expressed in degrees for this
    > example,
    > and North and West are positive.
    >
    > SF:
    > lat1 = 37 degrees
    > lon1 = 122 degrees
    >
    > SLC:
    > lat2 = 40 degrees
    > lon2 = 112 degrees
    >
    > So,
    >
    > a is the co-latitude of lat1, or 90-lat1 or 53
    > degrees.
    > b is the co-latitude of lat2, or 90-lat2 or 50
    > degrees.
    > ab is the difference of the longitudes, or
    > lon1-lon2 or 10 degrees.
    >
    > Solving for c one learns the distance in
    > degrees between SF and SLC,
    > which is about 8.375 degrees.  To get a
    > distance we understand,
    > multiply degrees times 60 nmi per deg and you
    > get 502.5 nmi.
    >
    > One then can re-substitute back in and use the
    > formula again to
    > determine the initial great circle course.  In
    > this case we
    > use a as co-latitude of lat1, b is the distance
    > we just
    > computed (8.375 degrees -- make sure to use
    > degrees), c is
    > the co-latitude of lat2, and the angle ab is
    > the initial
    > course.  We have rotated the whole spherical
    > triangle around
    > to put the sides a,b, and c where we know their
    > values,
    > leaving ab to be solved for.  Substituting and
    > solving one
    > determines that the initial course is 65.9588
    > degrees.
    >
    > ---
    >
    > In sight reduction:
    >
    > a is the co-latitude of the assumed position
    > b is the declination of the body (say the sun)
    > ab is the hour angle of the body
    > c is the altitude of the body
    >
    > ---
    >
    > Note too that there are equivalent ways of
    > writing the spherical
    > law of cosines.  One can write it as
    >
    > cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
    >
    > where the pairs of sin/cos of a/b are switched.
    >  This allows one to do
    > great circle calculations using latitudes
    > directly,
    > without using co-latitudes.  The origin is
    > moved from the pole to
    > the equator, so to speak.  This form is often
    > handier but the first
    > version is the canonical version.
    >
    > Hope this helps.
    >
    > Dan
    >
    
    
    
    Dave Weilacher
    .US Coast Guard licensed captain
    .    #889968
    .ASA certified sailing and celestial
    .    navigation instructor #990800
    .IBM AS400 RPG contract programmer
    
    
    

       
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