The original question:
Distribute N points according to a homogeneous Poisson point process in a two-dimensional space of unit area.
That doesn't make sense if N is a fixed number. For the definition of the Poisson point process, the N has to be a Poisson random variable with its mean related to the area/size of the simulation region. This is non-negotiable. But if you fix N=n to some natural number (that is, in probability language, you condition on N=n), you then get a binomial point process.
As Nguyen Anh suggested above (and I wrote in another response), 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 blogged about simulating these two point processes. The binomial point process on a rectangle is here. The Poisson point process on a rectangle and on a disk.
The code for a binomial process on a disk would be.