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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?

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We need an approximation for computing the circumference of a circle too, as $\pi$ isn't rational... –  J. M. Oct 18 '11 at 0:04

2 Answers 2

up vote 5 down vote accepted

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

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Indeed, I believe the reason the "elliptic integrals" were originally given that name is this problem. Arc length for an ellipse was--at least at that time--the best-known example of such an integral. –  GEdgar Oct 21 '11 at 13:35
    
For the series written down, see "Circumference" here en.wikipedia.org/wiki/… –  GEdgar Oct 21 '11 at 13:37

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
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