Formula used to calculate Perimeter of an Ellipse?

Question

aaronandyson · Answer

@freckles

directrix · Answer

Do some reading here. It appears that there are several. https://www.mathsisfun.com/geometry/ellipse-perimeter.html

aaronandyson · Answer

After reading over there,I've made this post. xD

empty · Answer

Here's a "closed form" in my opinion that's fantastic: http://www.had2know.com/academics/arithmetic-geometric-mean-calculator.html The problem is I'm not sure if we can manipulate the integral for arclength of an ellipse parametrized by: $$r(t) = \langle a \cos t , b \sin t \rangle $$ $$L = 4 \int_0^{\pi/2} \sqrt{a^2 \sin^2 t + b^2 \cos^2 t} dt$$ into an integral of the form: $$K(z) = \int_0^{\pi/2} \frac{d \theta}{\sqrt{1 - z^2 \sin ^2 \theta}}$$ If you can do that, you can get something that's just as good as writing $\sqrt{2}$ or $\ln 2$ since the AGM function converges super rapidly, for instance look here: https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm "only 25 iterations producing 45 million correct digits of π"