Circle Point Picking
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).
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