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Ball Point Picking


Ball point picking is the selection of points randomly placed inside a ball. n random points can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n].

Pick variates X_1, ..., X_n independently from a standard normal distribution and variate Y independently from an exponential distribution with parameter lambda=1. Then the distribution of points

 ((X_1,...,X_n))/(sqrt(Y+sum_(i=1)^(n)X_i^2))
(1)

is uniform over the unit n-ball. Note however that in practice, computation of N points inside a unit n-ball may still be slower using this technique than computing additional points (e.g., by a factor of  approx 2^2/pi for n=2 and  approx 2^3/(4pi/3) for n=3) inside an n-cube of edge length 2 and discarding the ones with norm greater than 1.

This result is a special case of a beautiful general result described by Barthe (et al. 2005) which may be stated as follows. For p>0 and a sequence of real numbers x={x_i}_(i=1)^infty, define the p-norm as

 |x|_p=(sum_(i=1)^infty|x_i|^p)^(1/p).
(2)

The space of all infinite sequences with |x|_p<infty is then denoted l_p, and the space R^n equipped with quasi-norm |·|_p is denoted l_p^n. Finally, the unit ball of l_p^n is defined as B_p^n={x in R^n;|x|_p<=1}.

Now, pick X_1, ..., X_n independently with probability density given by

 P_p(x)=(e^(-|x|^p))/(2Gamma(1+p^(-1))),
(3)

where Gamma(x) is the gamma function, and Y an independent random variate from an exponential distribution with mean 1. Then the random vector

 ((X_1,...,X_n))/((Y+sum_(i=1)^(n)|X_i|^p)^(1/p))
(4)

is uniformly distributed over the unit ball of l_p^n (Barthe et al. 2005).


See also

Ball Line Picking, Disk Point Picking, Noise Sphere, Sphere Point Picking

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References

Barthe, F.; Guedon, O.; Mendelson, S.; and Naor, A. "A Probabilistic Approach to the Geometry of the l_p^n-Ball." Ann. Probab. 33, 480-513, 2005.

Referenced on Wolfram|Alpha

Ball Point Picking

Cite this as:

Weisstein, Eric W. "Ball Point Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BallPointPicking.html

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