Random points on a unit sphere without biasing 2005-06-14 - By Andy Jones
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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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