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 not-too-weird shape, e.g. a square), 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 maybe 4 decimal places, and it wasn't any number I recognized.

EDIT: I originally wrote that I remembered computing the number to "perhaps 5 or 6 decimal places", but now I'm thinking that's unlikely. Changed to "maybe 4 decimal places".