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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...2011-10-18

2 Answers 2

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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.

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    For the series written down, see "Circumference" here http://en.wikipedia.org/wiki/Ellipse#Mathematical_definitions_and_properties2011-10-21
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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