Location of highest density on a sphere

Question

I have a lot of points on the surface of the sphere. How can I calculate the area/spot of the sphere that has the largest point density? I need this to be done very fast. If

Answer 1

Partition the sphere into equal-area regions (bounded by parallels and meridians) as described in my answer there and count the points in each region.

The aspect ratio of the regions will not be uniform (the equatorial regions will be more "squarish" when N~M, while the polar regions will be more elongated). This is not a problem because the diameters of the regions go to 0 as N and M increase. The computational simplicity of this method trumps the better uniformity of domains in the other excellent answers which contain beautiful pictures.

One simple modification would be to add two "polar cap" regions to the N*M regions described in the linked answer to improve the numeric stability (when the point is very close to a pole, its longitude is not well defined). This way the aspect ratio of the regions is bounded.