Need surface area of N-Ellipsoid.

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Discussion Overview

The discussion revolves around the surface area of an N-dimensional ellipsoid, exploring mathematical approaches and integrals involved in calculating this surface area. Participants share their attempts and challenges in deriving the formula, as well as related concepts such as the volume of ellipsoids.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant inquires about the surface area of an N-dimensional ellipsoid and seeks assistance.
  • Another suggests using a linear substitution to relate the integral to the surface area of an N-sphere.
  • A participant expresses difficulty in integrating the proposed approach and shares their integral setup involving axis lengths.
  • There is a discussion about the volume of the ellipsoid, with one participant stating they derived it using a similar method.
  • Some participants propose that the surface area can be expressed in terms of the surface area of a unit ball, but this is challenged by another participant who points out inconsistencies in this reasoning.
  • One participant notes that the surface area of an N-ellipsoid may not yield an easily integrable expression and references the use of incomplete elliptic integrals for the 2-ellipsoid case.
  • Another participant questions whether their setup for the integral was correct, indicating uncertainty in their approach.
  • There is acknowledgment that the surface area for N = 3 involves complex formulas, including elliptic integrals.

Areas of Agreement / Disagreement

Participants express differing views on the integrability of the surface area of an N-ellipsoid, with some suggesting it can be derived from simpler forms while others argue that it is inherently complex. No consensus is reached on the correct approach or formula.

Contextual Notes

Participants mention the potential complications in deriving the surface area and volume, including the need for elliptic integrals and the challenges of coordinate transformations. Specific assumptions about the ellipsoid's axes and their implications for the surface area calculation are also noted.

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


tiny-tim said:
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
 
Show us. :smile:
 


tiny-tim said:
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
 


tiny-tim said:
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)
 
Hi tohauz! :smile:

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

and the plural of "axis" is "axes" :wink:)
tohauz said:
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:
 
tiny-tim said:
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.
 
tohauz said:
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
tiny-tim said:
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
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
Mute said:
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
tiny-tim said:
oops! :redface:

should have been (a1*...aN)(N-1)/N*surfacearea(unit ball) :blushing:

Could you please tell me how you got it?
 
  • #14
tohauz said:
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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