Newsgroups: rec.puzzles
From: Dan Hoey <haoyuep@aol.com>
Date: Thu, 13 Dec 2007 17:45:33 -0500
Subject: Re: Uniform Distribution of Points on The Surface of a Sphere

Bill wrote:
> What is meant by the "uniform distribution of points on the surface of
> a sphere"?

> IMHO,  it means that every great circle contains as many points as
> every other great circle.

I take it you want to define what characteristic of a finite set of
points on the sphere would constitute uniformity.

Your criterion does not make any sense I can see as written, as we can
certainly find great circles that do not contain any of the points.
One might interpret your condition as requiring that every _hemisphere_
contains as many points as any other--that is, that every great circle
divides the set in half--but that would not make for a good criterion.
We could place all our points in the arctic and antarctic and still
satisfy that criterion.

The usual criterion for uniformity (on the sphere, as in any space of
finite measure) is that every _small_ disc contain a number of points
proportional to its area.  This is not possible to achieve exactly,
but can be approximated when as the number of points gets large.  Two
common methods are Monte Carlo and Bucky Fuller.

Dan