(x,y) = i * u1 + j * u2
u1 = (1,0)
u2 = (1/2, sqrt(3)/2)
If you want to generate all centers in the hexagon (not just the triangle) and still preserve the 1:1 nature of the map from (i,j) to (x,y), you could just rotate 6 times by 60 degrees each time.
Doing just this would introduce duplicates along the edges, where the triangular segments abut each other. You could eliminate those by insisting that i>0 rather than just i>=0.
You could get the same effect as rotation by introducing a third basis vector u3, but then there would be duplicates in the map from (i,j,k) -> (x,y).
As mentioned in the post - deduping is cheap and easy so introducing the third basis vector u3 could potentially be a great move away from the recursive logic that currently plagues the project.
The packing density is about aligning on a hexagonal lattice, the points of which are a simple function of the inner circle's radius - the points in each row are 2r apart and every other row is horizontally offset by 2r * tan(60) (I hope that's right). Could you not build a row of circles out from the center to the right edge and copy it to the rows above, offsetting as required? The length of any given row is then half of the chord it sits on i.e. sqrt(r^2 - d^2) where d is the distance from the center, in this case 2r * row-index -- with this you can then filter out which circles are outside the containing circle for that row. This fills the top-right quarter segment of the circle, and the whole thing can be filled by flipping the coordinates along x to get the bottom right, and flipping all of that along y to get the other half, then dedup the cross that forms in the middle.
I think, in terms of coverage, this must always be at least as good as a hexagon built out from the center, but that should be a rare case and most of the time it'd be better.
Evidently I'm not a mathematician. Am I being stupid here?
I can believe it for restaurants, hotels and popular hang-outs. But thinking about churches, strip clubs, the urologist, etc., I wonder if there are "blind spots" or other areas of low-quality data. Or even areas of poor cell coverage?
Maybe in most cities there are enough other places around that people frequent to cover over any blind spots? I'd be interested to know whether they've found any places like that?
I think a 'real' analysis of the data at a much larger scale (probably state-wide would be a good test) would yield some interesting data on the effectiveness of this implementation in specific.
As mentioned in the post - the circle packing implementation itself is general enough to be used in other applications. The point was to show off a cool potential application for a solution to a non-intuitive problem.