Create Random Number Within An Annulus

I am trying generate a random number that is within an annulus, i.e. we have a max and min radius. I tried doing: while True: x=random.uniform(-maxR, maxR) y=random.uniform

Solution 1:

In general you can either draw the correct distribution directly or use rejection.

To draw directly use

• draw theta uniformly on [0,2pi): theta = random.uniform(0,2*pi)

• draw r from the power-law distribution r^1.

  The only complexity compared to doing this for a circle is that you PDF runs from [r_min,r_max] not [0,r_max]. This leads to

  CDF = A \int_{r_min}^{r} r' dr' = A (r^2 - r_min^2)/2

  for A the normalizing constant

  A = 2/(r_max*r_max - r_min*r_min)
  

  implying that

  r = sqrt(2*random.uniform(0,1)/A + r_min*r_min)
  

  and you can simplify slightly.

• then compute (x,y) by the usual transformation from radial coordinates x = r * cos(theta)y = r * sin(theta)

This method of integrating the PDF, normalizing the CDF and inverting is general and is sometimes called the "Fundamental Theorem of Sampling".

Rejection

Draw (x,y) on a box big enough to contain the annulus, then reject all cases where `r = sqrt(xx + yy) exceeds r_max or is less than r_min.

This is reasonably efficient if the hole in the middle is small, and very inefficient if the hole is large.