From casida@CHIMCN.UMontreal.CA  Sun Dec  6 06:01:10 1992
From: casida@CHIMCN.UMontreal.CA (Casida Mark)
Message-Id: <9212061601.AA07791@chims1.CHIMCN.UMontreal.CA>
Subject: uniform random distribution on sphere
To: chemistry@ccl.net
Date: Sun, 6 Dec 92 11:01:10 EST


Since the following question seems to have stimulated a significant
amount of interest on the list, I thought I'd just mention an old
trick I happen to know for solving the classic problem of generating uniformly
distributed random points on the surface of a sphere.  I recall getting
this from the (math?) literature many years ago, but have lost the
reference.

> Hi fellow chemists,
> This may fit better in comp.graphics, but someone in this group
> may have experienced it already. So here it goes.
> I want to generate points on the surface of a sphere as randomly
> as possible. These points may be used for drawing van der Waals
> surface of an atom. I tried with symmetrically distributed points
> (generated by varying phi and theta, etc), but due to symmetry,
> the points looked like marching ants at certain angles.
> Since many of symmetrically distributed points are likely to be
> overlapped when they are placed in 2-D screen, increasing the
> density of VDW points would require unnecessarily large number
> of sphere points. Random distribution of points would help
> minimizing the number of points required and will give a better
> appearance.
> Any ideas?
> -DCL
> lim@rani.chem.yale.edu

The best solution I know (and one which works well when adopted to 
N-dimensions!) is to generate (x,y,z) points with a Gaussian random 
distribution in each coordinate.  This gives you a spherically 
symmetric 3-dimensional distribution which you can then project 
radially onto the surface of a sphere of radius R:
  
           r^2 = x^2 + y^2 + z^2
  
           (x,y,z) -> (R/r)*(x,y,z)
  
Gaussian random number generators are widely available.
  
                               Mark E. Casida
                               casida@chimcn.umontreal.ca