One can sample coordinates of the surface of a unit radius $n$-dimensional sphere uniformly using the following method: independently generate a vector of $n$ standard normal random variables $\mathbf{x}=[X_1,\ldots,X_n]$, compute the Euclidean norm of this vector $\|\mathbf{x}\|_2=\sqrt{X_1^2+\ldots+X_n^2}$, then vector $\mathbf{x}/\|\mathbf{x}\|_2$ will contain the uniform sample of a coordinate from the surface of a unit-radius $n$-sphere.

I am wondering if that method can be extended to sampling from an $n$-sphere with arbitrary radius $r>0$. My intuition tells me that multiplying the sample from unit-radius $n$-sphere by $r$ might work, but I haven't seen that mentioned anywhere, which makes me think that either my intuition is wrong or this may be a question with an obvious answer.

So, can someone clarify: is vector $r\mathbf{x}/\|\mathbf{x}\|_2$ a uniform sample from the surface of an $n$-sphere of radius $r$? If it is not, is there a way to use i.i.d. normal variates to obtain such a sample?

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    $\begingroup$ Yes. ${}{}{}{}$ $\endgroup$ – Rahul Apr 22 '13 at 8:35

Answer: "Yes." – Rahul Narain Apr 22 at 8:35

Additional remark: if you manage to take uniform samples from the unit sphere by whatever method, multiplication by $r$ will do the same for the sphere by radius $r$. This is because the pushforward of the normalized surface measure on the unit sphere by the map $x\mapsto rx$ is the normalized surface measure on the sphere of radius $r$.


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