 | | | Random points on a unit sphere without biasing | Random points on a unit sphere without biasing 2005-06-14 - By kim aldis
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Why Gaussian? Wouldn't normalising x/y/z random numbers work?
> -- --Original Message-- --
> From: owner-xsi@(protected)
> [mailto:owner-xsi@(protected)] On Behalf Of Alan Jones
> Sent: 14 June 2005 21:42
> To: XSI@(protected)
> Subject: Re: Random points on a unit sphere without biasing
>
> 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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