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I'm sure this is a simple problem, but I'm not quite sure if I'm doing this right.

Let me preface this with I am a computer-science person, and will need a tractable solution that I can program.

In most programming languages, you have a function rand() that will return a real number from 0 to 1. This is a uniform distribution.

However, I am trying to create a galaxy of stars. Thus they will be more dense toward the center, and less dense in the outer reaches of the galaxy.

I also want to set the maximum densities toward the center and minimum densities toward the outside edge. So for example, 5% density at the outer edge, and 60% density at the inner edge. And in between would be some scale of the cdf of the normal. (From which I will get and use the equation from wiki here.)

So I would do this <normal> = f(rand(), edgeDensity, coreDensity)

I was thinking of taking $coreDensity - edgeDensity$ and adjust the scale of the normal distribution to be

$f(x, e, c) = [\phi(x) \cdot (c-e)] + e$,

where we have the following:

  • $0 \leq e < c \leq 1$

  • $0 \leq x \leq 1$

  • $\displaystyle \phi(x) = \frac{1}{2}\left[1+\mathrm{erf}\left(\frac{x-\mu}{(2\sigma^2)^{1/2}}\right)\right]$

But this is where I get stuck. I'm not sure which values I should use for $\sigma$ and $\mu$ as well as I know $\mathrm{erf}$ is some error function, but is there a simple non integral solution to it?

Or is there a better way to do this?

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Nope, there is no simple closed form. Liouville proved this in the early-to-mid 1800's. There are very good numerical approximations, though. For starting points, see here and here, and, especially, W. J. Cody's paper here. – cardinal May 3 '11 at 22:27
what is the $R_{lm}()$ function in Cody's paper. – ohmusama May 3 '11 at 23:28
A rational function of degree $\ell$ and $m$ in the numerator and denominator, respectively, as stated at the bottom of the first page. Coefficients for the rational functions are given on the subsequent pages. – cardinal May 4 '11 at 0:06
So it looks like the $$R_{lm}() = -100 \log_{10} max\Bigl(\dfrac{f(x) - f_{lm}(x)}{f(x)}\Bigr)$$ but it says that $f(x) = erf(x)$ which would imply a recursive element? or am I reading this wrong? – ohmusama May 4 '11 at 0:51
For generating normal random variates: Box-Muller transform. It's not the fastest algorithm out there (cf. ziggurat), but it's still quite fast and very easy to implement. – cardinal May 4 '11 at 3:01

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