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Re: NOTE SUBMISSION
Posted:
Feb 8, 2009 5:55 AM


In article <1810883.1233935248698.JavaMail.jakarta@nitrogen.mathforum.org>, Shahram Zafary <shahram_zafary@yahoo.com> 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.
ArithmeticGeometric 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{1k^2.sin^2 u} = pi / (2.M(1, k'))
E(k) = int_0^{pi/2} sqrt{1k^2.sin^2 u} du = (1  S) K(k)
where S = sum_n 2^{n1}.(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 46 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



