Date: Dec 26, 2011 10:37 PM
Author: thomasinventions@yahoo.com
Subject: Re: A single term formula for calculating the circumference of ellipse

That's fabulous.

Note, as I've used recently in a different fashion, (1-h) could be injected to improve upon the simplistic model as in:

C~pi(a+b)(4/pi)^[h+h^3(1-h)^5]

From this modest treatment, the maximum relative error goes from ~1.3E-3 to ~5.7E-4 and if altering the integral powers 3 & 5 to 2.688 & 4.952, there's a further reduction to ~4.4E-4.

Or, more in it's original form,

C~pi(a+b)(4/pi)^(h^0.991)

gets the maximum relative error close to ~4.8E-4. 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 ~3.6E-4:

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