The question of generating uniformly distributed points on the surface of n-dimensional unit ball has been already posted here a dozen of times. What I'm interested in is the proper scaling when we want to generalize the method to uniform distribution on the surface of non-unit radius n-ball. Specifically, I'm looking for the scaling for the "projecting Gaussian on the ball" method.

I know this is a fairly easy question, but I cannot figure out the math and I appreciate any comments or references.

  • 2
    $\begingroup$ Each point on the unit ball is, by definition, a unit vector. Multiplying all its coordinates by the intended radius accomplishes the desired rescaling: it's just a change in the units of measurement. $\endgroup$ – whuber Apr 4 '13 at 22:38
  • $\begingroup$ @whuber: so, the number of dimensions has no effect? How can one ensure that a simple linear scaling preserves the uniformity? $\endgroup$ – Ebrahim Apr 4 '13 at 22:57
  • $\begingroup$ "Uniform" on the sphere $S^{n-1}$ means the probability of any subset $E$ equals its $n-1$-volume divided by the $n-1$ volume of $S^{n-1}$. When you rescale the sphere by a nonzero factor $\rho$, all $n-1$-volumes are multiplied by $\rho^{n-1}$. Therefore the ratio of volumes of $E$ to $S^{n-1}$ remains unchanged. This argument applies to any distribution on any measurable subset of a Euclidean space, not just to surfaces of balls. $\endgroup$ – whuber Apr 5 '13 at 12:51

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