A uniform distribution of points on the circumference of a circle can be obtained by picking a random real number
between 0 and .
Picking random points on a circle is therefore a great deal more straightforward
than sphere point picking.

have the desired distribution (von Neumann 1951, Cook 1957). This method can also be extended to sphere point picking (Cook
1957). The plots above show the distribution of points for 50, 100, and 500 initial
points (where the counts refer to the number of points before throwing away).

Cook, J. M. "Technical Notes and Short Papers: Rational Formulae for the Production of a Spherically Symmetric Probability Distribution."
Math. Tables Aids Comput.11, 81-82, 1957.von Neumann,
J. "Various Techniques Used in Connection with Random Digits." NBS Appl.
Math. Ser., No. 12. Washington, DC: U.S. Government Printing Office,
pp. 36-38, 1951.Watson, G. S. and Williams, E. J. "On
the Construction of Significance Tests on the Circle and Sphere." Biometrika43,
344-352, 1956.