# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 * ========================================================