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    Re: Location Center of Responses thus Far
    From: R.H. van Gent
    Date: 2001 Jul 24, 10:10 AM

    George Huxtable wrote:
    
    > I have been pondering on what would be a meaningful way to define a centre
    > of gravity for individuals members scattered over the surface of a sphere.
    >
    > Somehow, I doubt whether averaging latitudes and then separately averaging
    > longitudes provides the best answer. It should be possible to do the job
    > properly, for anyone with a bit of time on his hands. Here is a suggested
    > mechanical analogy.
    >
    > Take a globe of the world, that will float. For example, I have a beachball
    > printed with the World on its surface, crudely marked with lat and long. If
    > you put it into a pool, it will float any way up, like most spheres.
    >
    > Now glue on to its surface a set of identical coins or weights, one for
    > each member, in the spot where they live, and chuck it into the pool again.
    > Now, the lowest point of this globe represents in some way a centre of
    > concentration of the members.
    >
    > It should be possible to devise a computer analogy to avoid having to do
    > the physical experiment. Would it produce the same answer as Dan Allen's
    > method? I wonder...
    
    This problem can be solved mathematically in the following way;
    
    Regard each position on the Earth's surface as a vector and reduce each
    longitude-latitude co-ordinate to its Cartesian co-ordinates (x,y,z):
    
      x = cos(lat) * cos(long)
      y = cos(lat) * sin(long)
      z = sin(lat)
    
    Add all the x's, y's and z's to form the sum vector (X,Y,Z), and perform
    the inverse operation:
    
      mean long = arctan(Y/X)
      mean lat  = arctan(Z/sqrt(X*X+Y*Y))
    
    When the arithmetical signs of X and Y are properly taken into account,
    there will be no longitude ambiguity (in FORTRAN one could use the
    ATAN2(X,Y) function to circumvent this).
    
    If I have some time later this evening, I will write a quick program and
    do some calculations.
    
    My lat-long = +52.086  -5.129
    
    ========================================================
    * Robert H. van Gent * Tel/Fax:  00-31-30-2720269      *
    * Zaagmolenkade 50   *                                 *
    * 3515 AE Utrecht    * E-mail: r.h.vangent{at}astro.uu.nl *
    * The Netherlands    *                                 *
    ********************************************************
    * Home page: http://www.phys.uu.nl/~vgent/homepage.htm *
    ========================================================
    

       
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