General Ellipsoid Area Formula: Detailed Explanation

[The Bill]

I'm looking for a source that fully derives the complete formula for the surface area of a general (triaxial) ellipsoid. I'd prefer a source that has more than just a full derivation, but also has a fair amount of prose discussion on this topic. Some historical context would be nice, as well. The sources I've seen so far just present the entire formula as a fait accompli. Which is fine for their intended audience of people who just need the answer for their engineering project or other work. But that's not me right now.

I wouldn't mind if the sources you can provide also discuss any of the various approximate formulas, but that's only of secondary interest to me right now.

 

[Baluncore]

I think you will need several references, that between them may just satisfy your want.

Encyclopedia of Mathematics and Its Applications 146.
George Dassios. 2012. Ellipsoidal Harmonics; Theory and Applications
Cambridge University Press. ISBN 978-0-521-11309-0

Start at page 265, and follow the references.

P.S. The last ref is;
I. Rivin. Surface area and other measures of ellipsoids. Advances in Applied Mathematics, 39:409–427, 2007. https://arxiv.org/pdf/math/0403375.pdf

 

[bob012345]

Here is a paper called On the Surface Area of the Ellipsoid by Stuart R. Keller

https://www.ams.org/journals/mcom/1...-1979-0514826-4/S0025-5718-1979-0514826-4.pdf