Random points on a unit sphere without biasing 2005-06-14 - By Andy Jones
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Is there a good way to restrict the sampling to within an angle of a
given vector with the Guassian method? To me, that and overall
simplicity seem to be the advantage of the first method on the mathwold
page.
-Andy
Alan Jones wrote:
>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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