- #1
tohauz
- 20
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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.
tiny-tim said: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}.tiny-tim said:Show us.
tiny-tim said:Show us.
tohauz said:Actually, I used same idea to find the volume and I got it:
it is a_{1}*...a_{N}*meas(unit ball)
tiny-tim said: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)
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.
tiny-tim said:oops!
should have been (a1*...aN)(N-1)/N*surfacearea(unit ball)
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
tiny-tim said:oops!
should have been (a1*...aN)(N-1)/N*surfacearea(unit ball)
tohauz said:Could you please tell me how you got it?
The surface area of a N-Ellipsoid is calculated using the following formula: S = 4π(a^p)(b^q)(c^r)Γ((p+1)/2)Γ((q+1)/2)Γ((r+1)/2)/Γ((p+q+r+3)/2) where a, b, and c are the semi-axes of the ellipsoid, and p, q, and r are the positive integers that determine the shape of the ellipsoid.
The surface area of a N-Ellipsoid refers to the total area of the outer surface of the ellipsoid, while the volume refers to the amount of space enclosed by the ellipsoid. The surface area and volume are calculated using different formulas.
Calculating the surface area of a N-Ellipsoid is important in various fields such as engineering, physics, and geology. It helps in understanding the shape and size of objects, determining the strength and stability of structures, and analyzing the physical properties of materials.
Yes, the surface area of a N-Ellipsoid can be approximated using different methods such as numerical integration, Monte Carlo simulations, or using the Gauss-Legendre quadrature formula. However, the accuracy of the approximation depends on the complexity of the ellipsoid's shape and the chosen method.
The surface area of a N-Ellipsoid is used in various real-world applications such as satellite communication, geodesy, and cartography. In satellite communication, the surface area is used to determine the amount of radiation that can be transmitted or received by the satellite. In geodesy and cartography, it is used to create accurate maps and models of the Earth's surface.