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Message Posted: Fri, 27 May 2005 @ 21:33:03 GMT

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Perhaps for your example they are but in general the formulae might give different results. "Law of Cosines for Spherical Trigonometry" (that's what the second formula in your posting is), despite its simpler look and mathematical exactness, is not recommended for *numerical* calculations.

At the risk of bothering the forum with the details I'll quote just a part from the GIS FAQ (the full version can be found on comp.inofsystems.gis):

"An UNRELIABLE way to calculate distance on a spherical Earth is the

Law of Cosines for Spherical Trigonometry

** NOT RECOMMENDED **

     a = sin(lat1) * sin(lat2)
     b = cos(lat1) * cos(lat2) * cos(lon2 - lon1)
     c = arccos(a + b)
     d = R * c

Although this formula is mathematically exact, it is unreliable for small distances because the inverse cosine is ill-conditioned. Sinnott (in the article cited above) offers the following table to illustrate the point:

     cos (5 degrees) = 0.996194698
     cos (1 degree) = 0.999847695
     cos (1 minute) = 0.9999999577
     cos (1 second) = 0.9999999999882
     cos (0.05 sec) = 0.999999999999971

A computer carrying seven significant figures cannot distinguish the cosines of any distances smaller than about one minute of arc. The function min(1,(a + b)) could replace (a + b) as the argument for the inverse cosine to guard against possible round-off errors, but doing so would be to "polish a cannonball". "

Regards,

Victor

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