 | | | Random points on a unit sphere without biasing | Random points on a unit sphere without biasing 2005-06-14 - By Andy Jones
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That's what the quasi-monte-carlo QMC algorithms do for you in Mental
Ray, right? mi_sample, for instance? I'm not sure what happens to the
even distribution once you pipe it through the Box-Muller transformation
or whatever you're doing, though. You could end up with some sort of
biasing or undesirable jitter patterns.
-Andy
OO's mailbot wrote:
> You are probably after an "evenly distributed" placement rather than
> truly random. You are likely to get better result faster by jittering
> an even distribution of sample rather than creating an even distribution.
>
> This is the way stocastic sampling works. rather than randomly picking
> points on a surface, you start by picking uniformaly distributed
> points and offset them slightly to break up the pattern.
>
> Most of what we call "random" in natural patterns are actually caused
> by cellular or molecular diffusion and are really not random at all
> but a stable system of attraction/repulsion. ie every point is spaced
> out more or less evenly across the surface, so that you never have
> points that are clustered too close or too far away from each other.
> With a very very high sample a random distribution will approximate
> that, but you have no controls over overlapping samples with a random
> number generator...
>
> oO
>
> On Jun 15, 2005, at 7:11 AM, kim aldis wrote:
>
>> True.
>>
>> A Gaussian distribution would concentrate the generated points
>> towards the
>> centre so you'd have fewer outside the sphere. Not perfect but a deal
>> better.
>>
>> <trivia>
>> Gaussian distribution just requires averaging a number of random
>> numbers.
>> The average of an infinite number of random numbers between lying
>> between -1
>> and 1 is zero. I once knew a Cambridge graduate mathematician who
>> couldn't
>> get that.
>> </trivia>
>>
>>
>>> -- --Original Message-- --
>>> From: owner-xsi@(protected)
>>> [mailto:owner-xsi@(protected)] On Behalf Of Alan Jones
>>> Sent: 14 June 2005 21:58
>>> To: XSI@(protected)
>>> Subject: Re: Random points on a unit sphere without biasing
>>>
>>> X, Y, Z ones would be the same as doing a cube getting more
>>> in the corners. Apparently gaussian is special... At least
>>> that's what I assume from what I read on the mathworld page.
>>>
>>> Cheers,
>>>
>>> Alan.
>>>
>>> On 6/14/05, kim aldis <kim@(protected)> wrote:
>>>
>>>> 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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