Overlaying a point cloud on a two-dimensional probability distribution such that the local point density corresponds to the local probability density
Overlaying a point cloud on a two-dimensional probability distribution such that the local point density corresponds to the local probability density is a good way to define the cell generators of a centroidal Voronoi tessellation (CVT).

CVTs don't loose their quantization precision by translation or rotation in concert with the underlying probability distribution. Not sure what your question is.