why we only have a approximation for every circumference for ellipse but not define a special formula for each ellipse

Question

Why do we only have an approximation for every circumference for ellipse, but we cannot define a special ratio formula for each ellipse? Is it possible for people to use a computer to find the exact "infinite series" relationship between the circumference of the ellipse and the major axis and minor axis?

Answer 1

This is the complete elliptic integral of the second kind. Series for it are well-known, and many numerical analysis packages or languages like Mathematica provide them as a function call.

Answer 2

Additionally, the algorithm for computing the circumference of an ellipse (based on the arithmetic-geometric mean) isn't too long if your environment doesn't support computing the complete elliptic integral of the second kind, $E(m)$:

ellipseCircumference[a_, b_] := Module[{f = 1, s, v = (1 + b/a)/2, w},
   w = (1 - (b/a)^2)/(4 v);
   s = v^2;
   While[True,
    v = (v + Sqrt[(v - w) (v + w)])/2;
    w = (w/2)^2/v;
    f *= 2; s -= f w^2;
    If[Abs[w] < 10^(-Precision[{a, b}]), Break[]];
    ];
   2 a Pi s/v
   ] /; Precision[{a, b}] < Infinity && Positive[a] && Positive[b]

(Yes, I'm using Mathematica. Yes, I know Mathematica has EllipticE[] available. I'm only using Mathematica for illustrative purposes. ;) )

Compare:

N[ellipseCircumference[3, 2], 20]
15.865439589290589791

With[{a = 3, b = 2}, N[4 a EllipticE[1 - (b/a)^2], 20]]
15.865439589290589791