Why map Hammersley 2D set's (u,v) to sphere's (θ, φ) coordinates (and not to (φ, θ) )? As explained in Holger Dammetz page Hammersley Points on the Hemisphere, the 2D coordinates (u,v) are usually mapped to the spherical coordinates (θ, φ).

As a result, the coordinate "uniformly distributed" - u in the 2D case - is the polar angle θ. This looks counter-intuitive to me, I would have mapped regularly spaced u values to regularly spaced azimuth φ values. To avoid "empty areas" like in this example:

which is generated by the mapping of the following 2D set:

What is the reason not to map (u, v) to (φ, θ) ?
You can of course, as you suggested, map (u, v) to (φ, θ). Unfortunately, it does not solve the problem for 5 points:



I've changed Holger Dammertz' code a bit (switched u and v), and you see that the problem still persists. For a higher number of points, it really doesn't seem to make a (visual) difference.

The Hammersley point set is a way to very quickly determine points on a sphere that have good, but don't need to have a perfect distribution. In the problematic case of very few points, Hammersley might just not be the tool for the job. Pre-calculating and storing points sounds like a better option to me.