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Replies: 8   Last Post: Apr 22, 2009 4:22 PM

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Michael Press

Posts: 2,137
Registered: 12/26/06
Posted: Feb 8, 2009 5:55 AM
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In article
Shahram Zafary <> wrote:

> Dear Sir;
> I have found out a single term formula for approximating the circumference of ellipse and I hope to print, and publish it. This formula is very simple and accurate.
> I will be grateful if you help and guide me about it.

Arithmetic-Geometric mean:

a_{n+1} = (a_n + b_n)/2
b_{n+1} = sqrt{a_n * b_n}
c_{n+1} = (a_n - b_n)/2
M(a, b) = lim_{n -> oo} a_n = \lim_{n -> oo} b_n

K(k) and E(k) are the complete elliptic integrals of the
first and second kind. k' = sqrt{1 - kk}.

K(k) = int_0^{pi/2} du / sqrt{1-k^2.sin^2 u}
= pi / (2.M(1, k'))

E(k) = int_0^{pi/2} sqrt{1-k^2.sin^2 u} du
= (1 - S) K(k)

where S = sum_n 2^{n-1}.(c_n)^2.

The perimeter, A, of an ellipse with semiaxes a and b,
0 < b <= a is given by

A = 4.a.E(k')
= (aa - S).2.pi/M(a, b)

where a_0 = a, b_0 = b, c_0 = aa - bb.

This will give the perimeter of the ellipse to 5 significant
figures with 4-6 iterations for e < .99, where e is the
eccentricity. The AGM converges quadratically so one more
iteration will give 10 significant figures.

So unless you know exactly how much accuracy you want at
compile time and are prepared to engineer approximations
for the full range of eccentricities that will be called
for, and absolutely need the speed of specialized
approximations, use the AGM as it will give answers with
accuracy specified at run time, and will not take much
time doing it.

Might as well code up the AGM and see how it performs anyway.

Michael Press

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