About the surface area of a prolate ellipsoid

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Discussion Overview

The discussion centers around the surface area of a prolate ellipsoid and its approximation in comparison to the surface area of a sphere. Participants explore the conditions under which the approximation can be valid, referencing specific equations from external sources.

Discussion Character

  • Technical explanation, Mathematical reasoning, Debate/contested

Main Points Raised

  • Some participants inquire about the limits for approximating the surface area of a prolate ellipsoid as 4πAB compared to the spherical surface area of 4πR².
  • One participant suggests substituting variables in the equations from a referenced source to explore approximations for small eccentricities.
  • Another participant expresses uncertainty about the functionality of the provided link and questions whether it leads to the result of 4πAC.
  • It is noted that the approximation cannot be exact and depends on the desired accuracy, with references made to specific equations for the surface area and the relationship between the radii.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity or limits of the approximation, and multiple viewpoints regarding the equations and their implications remain present.

Contextual Notes

Participants reference specific equations and relationships but do not resolve the mathematical steps or assumptions underlying the approximations discussed.

Tahmeed
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Is there any limit for which we can approximately write the surface area of a prolate ellipsoid to be 4piA*B comparing with the spherical 4piR*R??
 
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Tahmeed said:
Is there any limit for which we can approximately write the surface area of a prolate ellipsoid to be 4piA*B comparing with the spherical 4piR*R??
Look at eqn 9 at http://mathworld.wolfram.com/ProlateSpheroid.html.
You could substitute for a in terms of c and e using eqn 8, then make an approximation for small e.
 
haruspex said:
Look at eqn 9 at http://mathworld.wolfram.com/ProlateSpheroid.html.
You could substitute for a in terms of c and e using eqn 8, then make an approximation for small e.
I don't know why but the link isn't working for me. However does it bring 4piAC as a result?
 
Tahmeed said:
I don't know why but the link isn't working for me. However does it bring 4piAC as a result?
It can't be exactly that, of course, so it depends how good an approximation you want.
The exact equations at that link are
Area = ##2\pi a^2+2\pi ac\sin^{-1}(e)/e##, eqn (9), and ##a^2=c^2(1-e^2)##, eqn 8
where the radii are a, a, c.
 
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