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Lets say a torus is given by its major and minor radius. How to pick a set of N random points on its surface, with equal distribution?

("equal distribution" = for any chosen subsurface of the torus with same area, probability of any number of points obtained by this method and belonging to that area is the same)

I tried applying some methods from related Wolfram web page, but did not succeed. (This web page is for the sphere case instead of torus.) I asked myself this question out of sheer curiosity. But now I don't know how to solve it. :(

VividD
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    Just a thought: two-angle parametrization reduces the problem to choosing point inside the unit square with some easy-to-compute distribution. IIRC there are some algorithms for doing that (albeit they usually require distribution subject to certain restrictions) - perhaps some would be applicable? – Marcin Łoś Apr 29 '14 at 22:08

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Let's say the torus has medium radius R and inner radius r (so from R-r to R+r)

You have to choose $\theta$ and $\phi$ at random, $\theta$ being anything random between 0 and $2\pi$ and $\phi$ being the inner angle with 0 -> R+r, $\pi$ -> R-r

If you choose $\phi$ with probability density f, then your surface has probability $f(\phi)d\phi d\theta$ and has surface $(R+r.cos(\phi)).r.d\phi d\theta$

The probability must be equal for all surfaces and be equal to 1 in total so $f(\phi)= \frac{(R+r.cos(\phi))}{\int R+r.cos \alpha d\alpha}$

Thomas
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