
Re: A single term formula for calculating the circumference of ellipse
Posted:
Dec 26, 2011 10:37 PM


That's fabulous.
Note, as I've used recently in a different fashion, (1h) could be injected to improve upon the simplistic model as in:
C~pi(a+b)(4/pi)^[h+h^3(1h)^5]
From this modest treatment, the maximum relative error goes from ~1.3E3 to ~5.7E4 and if altering the integral powers 3 & 5 to 2.688 & 4.952, there's a further reduction to ~4.4E4.
Or, more in it's original form,
C~pi(a+b)(4/pi)^(h^0.991)
gets the maximum relative error close to ~4.8E4. There are two trouble areas, around h=0.3 and h=0.9.
I noticed I could play with the b term in the formula and improve the RE to ~3.6E4:
C~a*pi(1+(b/a)^(0.996718))(4/pi)^h
However, this improvement doesn't look as neat as the first formula shown herein.
Thomas Blankenhorn

