Non-random uniform disk sampling

See [1] [2] and there is a great treatment of the topic in this book [3]. I don&#x27;t know exactly what&#x27;s he doing, but if I was doing some kind of monte carlo integration (like the family of energy surfaces in 6-dimensional space that I plotted the volume of as a function of energy for my PhD thesis) I&#x27;d be really concerned about the &quot;uniform&quot; part.<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Low-discrepancy_sequence" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Low-discrepancy_sequence</a><p>[2] <a href="https:&#x2F;&#x2F;extremelearning.com.au&#x2F;unreasonable-effectiveness-of-quasirandom-sequences&#x2F;" rel="nofollow">https:&#x2F;&#x2F;extremelearning.com.au&#x2F;unreasonable-effectiveness-of...</a><p>[3] <a href="https:&#x2F;&#x2F;numerical.recipes&#x2F;" rel="nofollow">https:&#x2F;&#x2F;numerical.recipes&#x2F;</a>

Having one point at the center does not feel intuitive. For example the N=19 feels like it would be more uniform as same as N=20 but without center point. Similarly N=3 could be just the points on the outer circle, similar to what N=4 is now.

This subject has been studied in depth before, hasn&#x27;t it? E.g. Shirley&#x27;s mapping, which can be combined with a low-discrepancy sequence such as Hammersley.<p>Here is an introduction for those curious: <a href="https:&#x2F;&#x2F;towardsdatascience.com&#x2F;unit-disk-uniform-sampling-91880f3740fa" rel="nofollow">https:&#x2F;&#x2F;towardsdatascience.com&#x2F;unit-disk-uniform-sampling-91...</a>

Another interesting method is Fibonacci Lattice. <a href="https:&#x2F;&#x2F;observablehq.com&#x2F;@meetamit&#x2F;fibonacci-lattices" rel="nofollow">https:&#x2F;&#x2F;observablehq.com&#x2F;@meetamit&#x2F;fibonacci-lattices</a>

Isn&#x27;t this a variation of a sphere-packing problem? We know an optimally dense packing for 2D spheres. Then vary the radius of what is being packed until N of them fit in the disk&#x27;s area.

A bit of a strange ad-hoc solution.<p>First as other pointed out, one can generate the Sobol sequence or similar, but even with a random sample one can generate the random numbers deterministically. I.e Sample r^2 ~ Uniform(0,r0^2) and angle from U(0,2*pi) and use the fixed seed...