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 2pi. Picking random points on a circle is therefore a great deal more straightforward than sphere point picking.

n 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 x_1, x_2 from a uniform distribution on (-1,1), and rejecting pairs with x_1^2+x_2^2>=1. From the remaining points, the double-angle formulas then imply that the points with Cartesian coordinates


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).

See also

Circle Covering by Arcs, Circle Lattice Points, Circle Line Picking, Disk Point Picking, Sphere Point Picking

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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." Biometrika 43, 344-352, 1956.

Referenced on Wolfram|Alpha

Circle Point Picking

Cite this as:

Weisstein, Eric W. "Circle Point Picking." From MathWorld--A Wolfram Web Resource.

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