Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within distance r from any previously chosen sample point, until no more choices are possible.

As the size of the original region goes to infinity (keeping some constant shape), the expected density (that is, number of sample points per unit area) produced by maximal Poisson disk sampling of radius 1 approaches a constant. What is that constant?

Conversely, given a target density, what value of r will produce samplings whose expected density is equal to the target? (As the region size approaches infinity, of course.)

I don't see any mention of this in the standard algorithmic references by Cook, Mitchell, Dunbar/Humphries, or Bridson.

Empirically, the constant in question is roughly .67, assuming the little python implementation of Bridson's algorithm that I just whipped up is correct. A few years ago I wrote an optimized implementation of the algorithm which produced the constant experimentally to perhaps 5 or 6 decimal places, and it wasn't any number I recognized.