01-23-2020, 01:40 PM
(01-21-2020 05:16 PM)Albert Chan Wrote: [ -> ]\(\large p(a,b) = 2× \left( p({a+b \over 2}, \sqrt{ab}) - {\pi a b \over AGM(a,b)}\right)\)
In other words, calculate ellipse perimeter from a much less eccentric ellipse.
We could even do it in a novel way, from the *more* eccentric ellipse.
Below, LHS is the more eccentric ellipse, with \({a_0 + b_0 \over 2} = a, \sqrt{a_0 b_0} = b\)
\( p(a_0,b_0) = 2× \left( p(a,b) - {\pi b^2 \over AGM(a,b)}\right)\)
Using my Casio FX-115MS, AGM2 method for Halley's comet, numbers from here
A = 2667950000
B = 678281900
C = A²
D = B²
E = 0.5
F=B-A : C=C-EF² : B=√AB : A=A+F/2 : E=2E
Quote:\(p(a,b) = {p(a_0,b_0) \over 2} + {\pi b^2 \over AGM(a,b)}\)
Keep pressing "=" until F=0 (so that A,B,C all converged)
\(\large\pi\)(C+D)/B → ellipse perimeter = 11,464,318,984.1 km
This was similar to my lua code, with C=(A²+B²)/2, E=0.25, so that converged C/B = "radius".