Need surface area of N-Ellipsoid.

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Hi guys.
What is surface area of N dimensional ellipsoid?
Any help is really appreciated.
 

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  • #2
tiny-tim
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Welcome to PF!

Hi tohauz! Welcome to PF! :smile:

Hint: use a linear substitution to turn the integral into the surface area of an N-sphere :wink:
 
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Hi tohauz! Welcome to PF! :smile:

Hint: use a linear substitution to turn the integral into the surface area of an N-sphere :wink:
I tried. But I'm getting something which is not easily integrable
 
  • #4
tiny-tim
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Show us. :smile:
 
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Show us. :smile:
Suppose that ellipsoid has axis a_{1},...,a_{N}.
Then S=[tex]\int_{{\sum\frac{x^{2}_{i}}{a^{2}_{i}}<1}[/tex]dS(x)-surface integral.
Then i solved for x_{N}=a_{N}([tex]\sqrt{1-(\frac{x_{1}}{a_{1}})^{2}-...-(\frac{x_{N-1}}{a_{N-1}})^{2}}[/tex]) and used that formula for evaluating the surface area, where you need evaluate N-1 dimensional volume integra (Found the partial derivatives of x_{N} w/r to x_{i} and etc). IN that integral i made a substitution , i.e. linear transformation x_{i}=y_{i}a_{i}, i=1,...N-1. I got:

Integral over {B(0,1)} of[tex] {\sqrt{1+((\frac{a_{N}}{a_{1}})^{2}-1)y^{2}_{1}+...+((\frac{a_{N}}{a_{N-1}})^{2}-1)y^{2}_{N-1}}} [/tex]*Jac(Transformation), where B(0,1) is n-1 unit ball.
I think i made a mistake, but i can't find it. Thanks and by the way, what is the good textbook to brush up on these things
 
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Show us. :smile:
Actually, I used same idea to find the volume and I got it:
it is a_{1}*...a_{N}*meas(unit ball)
 
  • #7
tiny-tim
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Hi tohauz! :smile:

(use the X2 tag just above the Reply box … a1

and the plural of "axis" is "axes" :wink:)
Actually, I used same idea to find the volume and I got it:
it is a_{1}*...a_{N}*meas(unit ball)
That's right! :smile:

And you can do the same thing for surface area …

a1*...aN*surfacearea(unit ball) :wink:
 
  • #8
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Hi tohauz! :smile:

(use the X2 tag just above the Reply box … a1

and the plural of "axis" is "axes" :wink:)


That's right! :smile:

And you can do the same thing for surface area …

a1*...aN*surfacearea(unit ball) :wink:
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.
 
  • #9
tiny-tim
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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! :redface:

should have been (a1*...aN)(N-1)/N*surfacearea(unit ball) :blushing:
 
  • #10
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oops! :redface:

should have been (a1*...aN)(N-1)/N*surfacearea(unit ball) :blushing:
OK. Where is my mistake? Thanks
i'm having trouble with finding it
 
Last edited:
  • #11
Mute
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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
 
  • #12
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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
I see what you are saying, but at least did I set it up correctly?
 
  • #13
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oops! :redface:

should have been (a1*...aN)(N-1)/N*surfacearea(unit ball) :blushing:
Could you please tell me how you got it?
 
  • #14
tiny-tim
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Could you please tell me how you got it?
Sorry :redface:, my formula seems to be wrong …

I thought it would just be a matter of changing the coordinates, and multiplying by the appropriate factors, but Mute's :smile: link makes it clear that that doesn't work, and that the surface area, even for N = 3, is a complicated formula using "elliptic integrals".
 

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