We are told (here, for example) that the circumference or perimeter of an ellipse can only be approximated without the use of integration. How could we use integration to calculate the exact circumference of an ellipse with semi minor and major axes $a$ and $b$, respectively? Is there a generalized formula?
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2Does this answer your question? why we only have a approximation for every circumference for ellipse but not define a special formula for each ellipse It is the first link stackexchange shows as related. – Ethan Bolker Feb 13 '23 at 16:07
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Not really. I'm not looking for a computer algorithm solution. – Feb 13 '23 at 16:10
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There is no formula (in the sense you seem to want) for the perimeter of an ellipse. That's why elliptic integrals must be evaluated numerically. Here's another duplicate also suggested by stackexchange: https://math.stackexchange.com/questions/2041110/is-there-a-simpler-way-of-finding-the-circumference-of-an-ellipse?rq=1 – Ethan Bolker Feb 13 '23 at 16:17
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Okay, thank you. – Feb 13 '23 at 16:23
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There are double/single series expansions for the elliptic integrals – Тyma Gaidash Feb 13 '23 at 17:04
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The answer by J. M. on the linked dupe target shows the algorithm to compute the ellipse perimeter via the AGM (Arithmetic-Geometric Mean), which converges very quickly. – PM 2Ring May 19 '23 at 00:34
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@PM2Ring I'll check it out, thanks. – May 19 '23 at 14:41