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I'm new to the world of ellipses and have come to understand that calculating the exact circumference of an ellipse is not a trivial matter that involves elliptic integrals and elliptic functions.

However, does elliptic integration produce the exact circumference, are they workable? Are there any known and proven exact ellipse measurements out there?

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    $\begingroup$ We can calculate the elliptic integral to arbitrary precision. This seems like a non-question. $\endgroup$ Sep 22, 2020 at 16:12
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    $\begingroup$ Trivially the ellipse with $a=b=1$ has circumference of $2\pi$. $\endgroup$ Sep 22, 2020 at 16:16
  • $\begingroup$ @GoodMorningCaptain Thanks I edited the question, what about $a=1$ and $b=2$? $\endgroup$
    – Babiker
    Sep 22, 2020 at 16:21
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    $\begingroup$ The ellipse with $a=1$ and $b=\frac{\sqrt3}{2}$ (and therefore eccentricity $\frac12$) has a circumference which we can write in terms of the gamma function, specifically $\Gamma(\frac34)$. But then the gamma function is not very nice, either $\endgroup$ Sep 22, 2020 at 17:15
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    $\begingroup$ I think @MishaLavrov ’s example with $\Gamma(\frac34)$ is as explicit as anybody could ask for. $\endgroup$
    – Lubin
    Sep 22, 2020 at 18:11

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