Although I prefer analytic forms for the circumference of the ellipse, at least I will provide a means for calculating the exact ellipse circumference to 15 significant digits using the arithmetic-geometric mean in direct fashion as opposed to the indirect forms I presented prior that required a simple calculation of the derivative of a specific agm function at a particular point. I did come up with an analytic form to effect 100 parts per trillion relative error and I would release it upon favorable circumstances.
For semi-minor axis, b, and semi-major axis a, the algorithm is as follows:
PA1=(a+b)/2 ; PG1 = sqrt(a*b)
PA2=(PA1+PG1)/2 ; PG2=sqrt(PA1*PG1)
PA3=(PA2+PG2)/2 ; PG3=sqrt(PA2*PG2)
...
Circumference = pi*[(a+b)^2-2(PA1-PG1)^2-4(PA2-PG2)^2-8(PA3-PG3)^2...-2^k(PAk-PGk)^2]/(PAk+PGk)
Now, upon infinite iterations (for practical purposes, in desiring 15 significant digits, only about 6 iterations are required when (a/b)<10^6) of the AGM for a,b, lim k->inf (PAk-PGk)=0 which essentially means the denominator converges to 2*agm(a,b).
The one limitation, besides not being analytic to easily be able to find the derivative, etc, is that for the degenerate case, the circumference upon infinite iterations converges to pi*a and not 4*a as desired. A simple if,then statement would provide for the exception.
More important than any of this interesting math is to get our world on track to prosperity. This is why I spent the last 6 years writing my book and many associated websites and tirelessly writing our failed leaders including over 100 times to the Federal Reserve to try to get them to stop the economic anti-stimulus QE.
Please read the book that I have at the lowest price available:
Thoughtful Living
and hundreds of extra pages from sites you can see from my profile here, with the two most important ones:
Thoughtful Living Book Extras and
Proposed solutions (Extension of last section of Thoughtful Living book)
