boost::uniform_on_sphere in not uniform on sphere

[usu_acm]

It's clear that your previous (1.29) and current verisons 
(1.32) has no really uniform point distribution on sphere. 

This fact can be proved by observation that 
distributions of all coordinates are independent on each 
other.

For example for 3D-sphere really uniform distribution is 
given by Euler angles (x, arcsin(2t - 1)), where x is 
uniformly distributed random value on [0..2*pi].

[nobody]

Logged In: NO 

it is correct that each coordinate is selected independently, 
but with a normal distribution.

(warning: the following is a little fast and loose)

for a normal distribution, the probability of landing at x will be 
a^(-b*x^2)

if you use the same distrubution for y and z, the probability of 
landing at (x,y,z) will thus be:
a^(-b*x^2) * a^(-b*y^2) * a^(-b*z^2) = 
a^(-b*(x^2+y^2+z^2))

you can recognize the latter as:
a^(-b*l^2), where l = length(x,y,z)

in other words, the probability of landing at (x,y,z) is only a 
function of the *length* of (x,y,z), i.e. it is unrelated to the 
*direction* of (x,y,z)

btw, mathworld has a great page on this subject: 
http://mathworld.wolfram.com/SpherePointPicking.html

/michael toksvig

[usu_acm]

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To: nobody.
Perfect! thnx a lot!