Recently I've been fascinated by the topic of perimeter/circumference of an ellipse. All the formulas I find are either approximations, or infinite series. Here's what I've been wondering:
- Is there any "compact" formula for the exact circumference of ellipse? What I mean by "compact" is that all of the infinite computations are factored out into constants, which can then be reused whenever computing a circumference again (similarly to how, for circles, all of the infinite calculations are "condensed" into the π constant).
- If the answer to 1. is "no", then is there any prove of this, or is it simply the case that we have not been able to find such an expression yet?
I've been trying to find some sources online for this, but I can't seem to find anything that either confirms, or denies, the existence of such an expression.
The circumference of the ellipse is given by $$P=4a \int_0^{\frac \pi 2}\sqrt{1-e^2 \sin ^2(\theta )}\,d\theta=4 a E\left(e^2\right)$$ where $e=\sqrt{1-\frac{b^2}{a^2}}$ is the eccentricity, and $E$ the complete elliptic integral of the second kind. This is only formula. Many approximations have been proposed; have a look here for a "few" of them.