Good point, but if you wanted to work with an arbitrary distribution in the plane as a generalisation of this I think you'd need to look for an equal-area map f

Good point, but if you wanted to work with an arbitrary distribution in the plane as a generalisation of this I think you'd need to look for an equal-area map from R2 to S2? (eg http://en.m.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection http://en.m.wikipedia.org/wiki/Lambert_azimuthal_equal-area_...) Edit: I mean R2 to S2 (although this is bijective)