ellipse circumference

Question

Here is a Wikipedia article about the circumference of an ellipse: http://en.wikipedia.org/wiki/Ellipse#Circumference

I don't know how Ramanujan developed the following approximation for the circumference of an ellipse:

$ \pi\left(3(a+b)-\sqrt{(3a+b)(a+3b)}\right) $

I don't know how to derive this approximation either: $\left(\text{where }h=\frac{(a-b)^2}{(a+b)^2}\right)$

$C\approx\pi\left(a+b\right)\left(1+\frac{3h}{10+\sqrt{4-3h}}\right)$

Can somebody explain how to derive the approximations for an ellipse's circumference? Also, can somebody prove that there is no closed/simple formula for the circumference of an ellipse?

Answer 1

Most approximations work best when the eccentricity $e:=\sqrt{a^2-b^2}/a$ (or your $h$) is $\ll1$. In this case the elliptic integral $E(e)$ can be developed into a series in terms of powers of $e$.

Nobody can memorize the coefficients of this series. Therefore it pays out to construct (by "reverse engineering") a simple function of $e$ (containing only fractions, square roots and the like) whose Taylor expansion coincides with the expansion of $E(e)$ for as many terms as possible. This is a problem to play with and has no unique best solution; see the formulas in the quoted Wikipedia article.

It is much more difficult to give approximation formulas that are good for large eccentricities, or even in the limit $e\to1-$, when the ellipse becomes terribly flat. The circumference is then not an analytic function of $\delta:=1-e$.

Answer 2

Early in undergrad I had a project on the circumference of an ellipse. You can find the writeup here. The result of the story is: it's really not easy. In there I noted that Ramanujan's approximation matches the first 9 terms of the Taylor expansion from the arc-length formula if you make the small eccentricity assumption (as noted by Blatter) (I wish I cited where I got this fact... silly undergrad me). So if you expand it all out, make that assumption, and drop everything after the 9th term you can probably simplify it to something close to the Ramanujan approximation.