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I know this question already asked and actually well explained in those answers, just one thing that actually confuses be in other answers what if i want to generate random points on the surface of n-Sphere with radius r uniformly?

As according to one of the algorithm, generate n-dimensional vector $[x_1,x_2,..,x_n]$, where $x_i$ is random numbers using Normal Distribution N(0,1) and then use following formula to generate random point on the surface of n-Sphere

$$\frac{1}{\sqrt{x_1^2+x_2^2+... +x_n^2}}[x_1,x_2,...,x_n]$$

I presume that this will generate random points uniformly on the surface of unit n-Sphere, so what if we want generate random points on the surface n-Sphere with arbitrary radius r do we need to change the variance in Normal distribution or what?

Please forgive me if its a stupid question.

Regards Ahsan

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  • $\begingroup$ Just multiply the whole vector by $r$. $\endgroup$
    – user856
    Jun 18 '14 at 23:31
  • $\begingroup$ Can u elaborate a bit please? $\endgroup$
    – Ahsan Iqbal
    Jun 19 '14 at 0:03
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    $\begingroup$ Take the random vector on the unit $n$-sphere, $\frac1{\sqrt{x_1^2+\cdots+x_n^2}}[x_1,\ldots,x_n]$, and multiply it by $r$. $\endgroup$
    – user856
    Jun 19 '14 at 2:19
  • $\begingroup$ Rahul is right. $\frac{r}{\sqrt{x_1^2+\cdots+x_n^2}}[x_1,\ldots,x_n]$ is a random vector on the $n$-sphere with radius $r$. $\endgroup$
    – mike
    Jun 19 '14 at 14:00
  • $\begingroup$ Please link the relevant related questions $\endgroup$
    – leonbloy
    Jun 19 '14 at 14:25
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From a comment by Rahul:

Take the random vector on the unit $n$-sphere, $$\frac{1}{\sqrt{x_1^2+...+x_n^2}}[x_1,...,x_n]$$ and multiply it by $r$.

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