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 *************************************************