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The implementation Robert van Gent suggests does not work in the general case.
This throughs some light on the adequance of  George Huxtable's beach ball model.

To sum up the method, the idea is to look for the center of gravity of all
positions in 3-dimensions and subsequently project it via a central projection
onto the surface of the Earth. This will fail if the center of gravity is in the
center of the Earth and hence in the center of projection. But there still may or
may not be a place to which the travel distances are minimized. This raises the
suspicion that George Huxtable's beach ball model is not equivalent to a solution
that minimizes either the average (= total) distance travelled by all
participants or, in the interest of fairness to all, the sum of the squares.

Let us look at two special examples and note that the beach ball with coins would
be in equilibrium in any position in both cases.

Assume one person on the south pole, another on the north pole. The two could
meet anywhere on the equator (for least square dist.) or anywhere at all (for
min. avg. dist.). The beach ball suggests the latter but not the former.

But now assume 3 persons on the equator at longitudes 0, 120 and 240. This time
there are exactly 3 solutions to both minimizing strategies, namely the 3
positions themselves. The beach ball does not suggest this at all.

The question therefore poses itself: Can it be shown at least for the non
degenerated case where the center of gravity of all positions is not in the
center of the Earth, that  the Huxtable/van Gent solution is minimal, and if so,
in which respect it would be so? Of course, there is always the tedious way of
filling a few pages of paper with algebraic scribble, but maybe somebody has a
Geistesblitz.

Best regards

Herbert Prinz (from 1368950/-4603950/4182550 ECEF)

P.S.

I must have anticipated this important problem. For quite a while now  I have
been using ECEF coordinates in my signature. This should simplify things
considerably, if you guys want to include me in your meeting and if we can get
the above method to work.



"R.H. van Gent" wrote:

> 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.

>
> 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).

   

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