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Sampling with multidimensional transformations

  1. Apr 18, 2013 #1
    I am not sure if I have the title right, but here is my problem:
    I have a ray which 'should be' shot vertically from a point p, but depending on the situation it can: 1) either be shot in any direction in the hemisphere above p 2) shot with an angle of no more than σ off the vertical 3) shot with an angle of no more than σ off the vertical by with a Gaussian distribution
    (See http://imgur.com/BMqWjoQ)
    http://imgur.com/BMqWjoQ

    First:
    I wish to generate a point uniformly distributed on a hemisphere. I did some derivations and I came up with:
    θ = acos(R1)
    ∅= 2∏R2
    x = sinθcos∅=cos(2∏R2)sqrt(1-R12)
    y = sinθsin∅=sin(2∏R2)sqrt(1-R12)
    z=cosθ=R1
    I confirmed this wit a text book, So Im pretty sure its right---
    Second:
    I want to generate a point uniformly but only within a small solid angle subtended by angle σ
    Similar derivation as before but the values for theta and phi are
    θ=acos(1-(1-cos(σ)*R1))
    ∅= 2∏R2
    Im pretty sure this is also right

    Third:
    (now this is where I need help)
    Instead of using a uniform distribution I would like to use a Gaussian distribution. I know Box Muller is one way of generating random number with a normal distribution (given a set of canonical numbers) but how do I use that now to generate ray directions that are normally distributed?

    Thanks for your help
     
  2. jcsd
  3. Apr 18, 2013 #2

    mfb

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    Staff: Mentor

    Can you do rejection sampling? The "best" implementation would depend on details of the parameters. "Use the second algorithm and reject the point with probablility 1-(value of gaussian)" should give reasonable results if the gaussian is not too narrow.

    Alternatively, generate a random number based on the modified distribution f(θ)=sin(θ)*gaussian
     
  4. Apr 18, 2013 #3
    What do you mean by "gaussian"? Is that one of my gaussian random numbers?
    I assume sin(theta) is measure of area
    Area = ∫0σ0sin(θ) Z1dθd∅

    (thanks for your reply)
     
  5. Apr 18, 2013 #4

    mfb

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    Staff: Mentor

    sin(θ) is (proportional to) the length of a circle around the vertical axis. It is just a weight for the gaussian distribution. I don't know if you want that gaussian as function of θ, or the projection on the floor, or whatever, just use what you like there. As function of θ, it gets $$f(\theta)=\sin(\theta)\exp\left(\frac{-\theta^2}{2\sigma^2}\right)$$
    Where σ is the width of the gaussian.
     
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