If the simulation windown/region has 1000 (or non random n) points, then it is not a Poisson process. It is a binomial point process, which happens to be also a homogeneous Poisson point process conditioned on having n points.
You treat this problem in polar coordinates. For each point, you uniformly choose a random angular coordinate (or component) on the interval . For the radial coordinate (or component), you first choose a random number on the unit interval , and then you --- and this is very important -- square root that random number and multiply it by the radius of the disk. You have to take the square root to assure uniform points (because the area of a disk/sector is proportional to the radius squared, not the radius). I recently answered this question in detail on my blog, where I discuss how to simulate a (homogeneous) Poisson point process on a disk. The code is here: