Re: [boost] Computing Random Points on Sphere

From: Lewis Hyatt (lhyatt_at_[hidden])
Date: 2007-06-05 16:53:15

Henrik Sundberg wrote:
> I doubt that the trig-version gives a good distribution:
> 
> inlineVectortrig(uniform_01<generator_type>&random) {
>    constfloatphi=2*M_PI*random();
>    constfloatcos_theta=2*random()-1;
>    constfloatsin_theta=sqrt(1-cos_theta*cos_theta);
>    returnVector(cos_theta,sin_theta*cos(phi),sin_theta *sin(phi));
> }
> 
> The result ought to be more dense at the poles.

The posted trig method is correct, it chooses a horizontal cutting plane 
uniformly. You can show that the surface area of a sphere contained 
between two z-coordinates only depends on the difference between the two 
coordinates, not their location, so this generates the correct distribution.

However, the "naive" method is incorrect, the resulting points are not 
uniformly distributed over the sphere. The correct version of the 
"naive" method is given here: 
http://mathworld.wolfram.com/SpherePointPicking.html

-Lewis