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Need surface area of N-Ellipsoid.

  1. May 20, 2009 #1
    Hi guys.
    What is surface area of N dimensional ellipsoid?
    Any help is really appreciated.
     
  2. jcsd
  3. May 21, 2009 #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:
     
  4. May 21, 2009 #3
    Re: Welcome to PF!

    I tried. But I'm getting something which is not easily integrable
     
  5. May 21, 2009 #4

    tiny-tim

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    Show us. :smile:
     
  6. May 21, 2009 #5
    Re: URGENT! Need surface area of N-Ellipsoid.

    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
     
  7. May 21, 2009 #6
    Re: URGENT! Need surface area of N-Ellipsoid.

    Actually, I used same idea to find the volume and I got it:
    it is a_{1}*...a_{N}*meas(unit ball)
     
  8. May 22, 2009 #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:)
    That's right! :smile:

    And you can do the same thing for surface area …

    a1*...aN*surfacearea(unit ball) :wink:
     
  9. May 22, 2009 #8
    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.
     
  10. May 22, 2009 #9

    tiny-tim

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    oops! :redface:

    should have been (a1*...aN)(N-1)/N*surfacearea(unit ball) :blushing:
     
  11. May 22, 2009 #10
    OK. Where is my mistake? Thanks
    i'm having trouble with finding it
     
    Last edited: May 22, 2009
  12. May 24, 2009 #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
     
  13. May 25, 2009 #12
    I see what you are saying, but at least did I set it up correctly?
     
  14. May 25, 2009 #13
    Could you please tell me how you got it?
     
  15. May 25, 2009 #14

    tiny-tim

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    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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