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Re: gsl_rand_dir_3d


On Mon, Aug 04, 2003 at 02:38:46PM -0600, James Theiler wrote:
> On Mon, 4 Aug 2003, Stuart O Anderson wrote:
> ] After looking at the algorith a while longer I've begun to suspect it has
> ] a more serious flaw than the s==0 condition.  If the Z component is
> ] uniformly distributed (since P(x*x+y*y < s | s<=1 ) = s) and the final
> ] vector's orientation in the X-Y plane is also uniform and independent of Z
> ] then the probability of selecting a final vector with -.5<Z<.5 is equal
> ] to the probability of selecting a final vector with -1<=Z<=-.5 or .5<=Z<=1.
> ] Since the surface area of these two sections is not equal but the
> ] probability of selecting a vector in each section is equal, the
> ] distribution cannot be uniform over the surface of the sphere, since the
> ] probability of selecting a vector in any two regions of equal surface
> ] area must be equal in a uniform distribution.

> Actually, I believe the surface areas of the two regions you describe
> *are* equal.  This is at first a little counter-intuitive, but you
> should work it out.  The region near the equator seems to have larger
> area because it is farther from the axis, but the region around the
> poles has an extra cosine effect which makes up for this.  Hard to
> describe without a picture, but if you try to work it out, I think
> you'll see what I'm talking about.

This is true.  A uniform Z from -1 to 1 is okay.  I haven't looked
into whether s is uniform, though.

I guess what I posted would be a trig method of figuring it out.
Or, if you *want* the random direction in spherical coordinates, it
is that (without psi).  Maybe it could be
gsl_rand_dir_3d_spherical() or something.  And/or something closer
to what I posted, gsl_rand_euler() which gives a random orientation
for a solid object.

If someone *really* wants to see it, I could post the math.

-- 
Jeff Spirko   spirko@lehigh.edu   spirko@yahoo.com   WD3V   |=>

The study of non-linear physics is like the study of non-elephant biology.

All theoretical chemistry is really physics;
and all theoretical chemists know it. -- Richard P. Feynman 


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