Disk Point Picking
CircularDistribution
To generate random points over the unit disk, it is incorrect to use two uniformly distributed variables r in [0,1] and theta in [0,2pi) and then take
x = rcostheta
(1)
y = rsintheta.
(2)
Because the area element is given by
| dA=2pirdr, |
(3)
|
this gives a concentration of points in the center (left figure above).
The correct transformation is instead given by
x = sqrt(r)costheta
(4)
y = sqrt(r)sintheta
(5)
(right figure above).
The probability function for distance d from the center of a point picked at random in a unit disk is
| P(d)=2d. |
(6)
|
The raw moments are therefore given by
| [画像: mu_n^'=2/(2+n), ] |
(7)
|
giving a mean distance of d^_=2/3.
See also
Circle Point Picking, Disk Line Picking, Point Picking, Sphere Point PickingExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Disk Point Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiskPointPicking.html