> Well, it is not easy to create such a continuously uniform distribution,
> uniform at a certain number is fairly simple on the other hand.

What is the problem you are adressing here?

My assumption (1-dimensional version): A sequence of numbers
between 0 and 1 such that for each initial segment of the sequence
the density is fairly even throughout the interval.

This can be done by choosing an irrational number r and taking the
sequence a(n)=(n*r) mod 1. The optimal r then is the golden ratio,
i.e. r=(sqrt(5)-1)/2.

Two dimensional version: here we take two such sequences, to two
irrational numbers r1 and r2. Those have to be independent in the
sense that 1, r1 and r2 are linear independent in the vector space
of the real numbers over the field of the rational numbers. While
it is easy to find such r1, r2, I do not know which give the best
results (in the same sense that the golden ration is optimal above).

The final version: So far we have a continuing uniformly distributed
sequence in the unit square. We have to map this to the hemisphere
without loosing uniformity. This can be achieved by

  (x,y) -> (sin(2pi*x)*sqrt(1-y^2),cos(2pi*x)*sqrt(1-y^2),y)

Regards,

  Mark

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