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Hi guys.
What is surface area of N dimensional ellipsoid?
Any help is really appreciated.
What is surface area of N dimensional ellipsoid?
Any help is really appreciated.
Hi tohauz! Welcome to PF!
Hint: use a linear substitution to turn the integral into the surface area of an N-sphere![]()
Suppose that ellipsoid has axis a_{1},...,a_{N}.Show us.![]()
Show us.![]()
Actually, I used same idea to find the volume and I got it:
it is a_{1}*...a_{N}*meas(unit ball)
Hi tohauz!
(use the X2 tag just above the Reply box … a1 …
and the plural of "axis" is "axes")
That's right!
And you can do the same thing for surface area …
a1*...aN*surfacearea(unit ball)![]()
That doesn't make a sense. Because, if a_{i}=r for all i's, we don't get the surface area ball with radiuis r.
oops!
should have been (a1*...aN)(N-1)/N*surfacearea(unit ball)![]()
I don't think you should expect to get something easily integrable for the surface area of an N-ellipsoid. In general the surface area of just a regular 2-ellipsoid is expressible in terms of incomplete elliptic integrals. Similarly, I don't think the 'circumference' of an ellipse has a nice expression in terms of elementary functions, either. (There are some closed-form special cases)
http://en.wikipedia.org/wiki/Ellipsoid#Surface_area
oops!
should have been (a1*...aN)(N-1)/N*surfacearea(unit ball)![]()
Could you please tell me how you got it?