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Re: Noise Sphere
*To*: mathgroup at smc.vnet.net
*Subject*: [mg25337] Re: Noise Sphere
*From*: Yossi Lonke <jrl16 at po.cwru.edu>
*Date*: Sat, 23 Sep 2000 03:36:02 -0400 (EDT)
*Organization*: Dept. Mathematics, CWRU
*References*: <8q785u$t6p@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Hello,
In a second thought, the explanation offered by mathworld is not
satisfactory. Think about it this way:
Draw a rectangle [0,2Pi] x [0, Pi], which is to be mapped onto the unit sphere
by the usual spherical coordinate transformation:
(theta, Phi) -> (sin(Phi) cos (theta), sin(Phi) sin (theta), cos(Phi) )
Now look at a piece of the rectangle above close to Phi = Pi, e.g, the
rectangle [0,2Pi] x [0,t]. Since cos (t) behaves like 1-t^2/2 near t=0,
a strip of width "t" in the domain gets mapped to a spherical cap of height of
approx t^2/2. (A picture is helpful, of course). Thus a uniform distribution
of points in the domain will be mapped in such a way that whatever was caputred
in
a strip of width t near Phi=0, will be confined to a spherical cap around
(0,0,1), whose
"height" is of the order of t^2 -- this explains why we see many points bunching
around (0,0,1).
Best Regards,
Yossi Lonke
Geoffrey Steeves wrote:
> : In studying whether a sequence of numbers is random, one can construct
> : what is called a Noise
> : Sphere(http://mathworld.wolfram.com/NoiseSphere.html). When I tried this
> : out, I got a non-uniform distibution of points concentrated along
> : the verticle axis (phi ~= 0) of the sphere. Initially I thought that this
> : was an artifact of my pseudo-random number generator, but I later tried
> : this out with "real" random numbers and got the same distribution.
> : Looking at the mapping(as it is shown on the web page):
>
> : theta = 2 * Pi * Random[]
>
> : phi = Pi * Random[]
>
> : r = Sqrt[Random[]]
>
> : I thought that the problem was with the distribution in phi. Is this a
> : mistake in the map? Or am I misunderstanding the what a Noise spehere is
> : supposed to be? Thanks for the help!
>
> : --
> : ___________________________________________________________________________
> : Geoff Steeves // University of Alberta Physics //
>
> : http://www.ualberta.ca/~gsteeves
> : ---------------------------------------------------------------------------
--
*************************************************
Dr. Yossi Lonke
Mathematics Department
Case Western Reserve University
10900 Euclid Avenue
Cleveland, Ohio 44106
216 368-5423
http://www.cwru.edu/artsci/math/lonke/home.html
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