Random points on a unit sphere without biasing 2005-06-14 - By Alan Jones
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Thanks Andy. After looking at the options I think I'm going to go with
the one at the bottom of the mathworld page. It suggested 3 gaussian
random numbers for X, Y and Z in a vector and normalizing the vector
(at least that's what I think it said from my limited math). I managed
to track down an apparently fast way to generate gaussian random
numbers from evenly distributed ones so with any luck I'm set.
Thanks again,
Alan.
On 6/14/05, Andy Jones <andy@(protected)> wrote:
> The mathworld article answers this perfectly. Basically, you pick one
> angle (theta) at random and a height (h) on the sphere at random. You
> can check this by calculating the approximated surface area (width *
> radius) of the bands around the sphere at different heights along the
> sphere, and take the limit as dh -> 0. At the poles, the width of the
> band is larger for a given height, and at the equator, the radius of the
> band is obviously larger. In the limit, they have a perfectly inverse
> relationship such that each band has the same area. More generally, a
> good way to randomly sample vectors within a given angle of an average
> vector is to restrict the height range to [cos(angle), 1].
>
> I think you can't pick points in a cube and normalize because you'll
> bias more points at the corners and fewer points in the middles of the
> faces, since a cube has more and less volume in those directions. The
> same problem occurs if you sample on a cube's surface area. The cube
> method is especially problematic because the biasing isn't determined by
> the original vector direction. Of course, that doesn't mean it doesn't
> work okay in practice for many applications.
>
> -Andy
>
> Alan Jones wrote:
>
> >Hi All,
> >
> >This is to the maths geniuses in the room. I want to generate X number
> >of points on a unit sphere, but be sure I won't have any biasing
> >involved.
> >
> >My first thought was just to use a couple of random numbers (let's
> >assume they don't have any bias) and then use those with a few sin and
> >cos function etc to generate the points. Though I have a feeling that
> >would give me more points around the poles.
> >
> >Anyone have some good suggestions?
> >
> >Cheers,
> >
> >Alan.
> >
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