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I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in $n$-dimensional space with n even, I am thinking in the phase space of a system with $f$-degrees of freedom that have dimensionality $2f,$ the aim is picking phase-space points that have the same total energy. Thanks in advance.

EDIT: OK, perhaps thinking in a phase space states with energy $E = T(p) +V(q),$ one can pick at random $q$ which energy is less or equal than $E,$ if the energy is less than $E$ the remaining momentum can be picked from the ball $T(p)=cte$. But this is not a satisfactory answer to the question I did. Is this approach correct? How to do it efficiently using a computer?

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  • $\begingroup$ The problem easily reduces to sampling from the surface of the unit sphere. For this, see stats.stackexchange.com/questions/7977 $\endgroup$
    – Steve Huntsman
    Commented Apr 12, 2017 at 15:10
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    $\begingroup$ @SteveHuntsman: May I ask, what is the reduction? $\endgroup$
    – Joseph O'Rourke
    Commented Apr 12, 2017 at 15:19
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    $\begingroup$ @JosephO'Rourke Determine an affine transformation mapping the sphere to the ellipsoid. Then sample uniformly from the sphere and apply the aforementioned affine transformation. By linearity (perhaps I should say affinity?), this yields a uniform sample from the ellipsoid. $\endgroup$
    – Steve Huntsman
    Commented Apr 12, 2017 at 17:03
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    $\begingroup$ @SteveHuntsman from the solid ellipsoid, yes, from the surface, no. $\endgroup$
    – Igor Rivin
    Commented Apr 12, 2017 at 17:23
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    $\begingroup$ @JosephO'Rourke I stand corrected! It is sort of funny that my original comment holds if the word "easily" is removed though. $\endgroup$
    – Steve Huntsman
    Commented Apr 12, 2017 at 19:38

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If your ellipsoid is not too squashed, the method described in the first answer to this MSE question should work decently. If it IS very squashed, it won't, but an ellispoid with $a \gg b > c> \dots $ is essentially a product of a lower dimensional ellipsoid with an interval, so you can reduce the problem to a lower dimensional one (with some (slight) loss of uniformity.)

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