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.
random points can be picked on a unit circle in the Wolfram
Language using the function RandomPoint[Circle[], n].
Random points on a circle can also be obtained by picking two numbers 
, 
from a uniform distribution
on 
,
and rejecting pairs with 
. From the remaining points, the double-angle
formulas then imply that the points with Cartesian
coordinates
 |  |  |
(1)
|
 |  |  |
(2)
|
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).
