
Re: A single term formula for calculating the circumference of ellipse
Posted:
Apr 27, 2009 8:22 AM


Shahram Zafary <shahram_zafary@yahoo.com> wrote: > My formula for calculating the circumference of ellipse is attached as a > small note. I will be grateful if you read and judge it. > > (See: > http://mathforum.org/kb/servlet/JiveServlet/download/12819218646683056551607/circumference%20of%20an%20ellipse.pdf)
Congratulations on finding a nice approximation! (Concerning your note, I should mention that your stated worst rel. error is overly pessimistic. Instead of saying rel. error < 0.0017, you should have said rel. error < 0.0013, but perhaps you simply made a typographical mistake.)
A few comments:
In your formula, (pi/4) is raised to the power 4 a b/(a + b)^2. It is interesting to note that that power can be written nicely in terms of the geometric and arithmetic means of the semiaxes' lengths, namely, it is the square of the ratio of those means: Letting gm = sqrt(a b) and am = (a + b)/2 for brevity, your power can then be written as (gm/am)^2. Your approximation formula for the perimeter of an ellipse could then be written as
4 (a + b) (pi/4)^((gm/am)^2)
But the fact that the ratio of means is squared is not crucial to the formula, and so it is natural to ask whether we could do "better" by raising gm/am to some power p other than 2.
1) If our objective is to minimize worst rel. error, then, by numerical methods, it can be determined that p = 2.016861... should be used. We then obtain rel. error < 0.00078 . (Of course, a disadvantage of using p = 2.016861... is that it's not as easily remembered as 2. But p could, if desired, be rounded to 2.017 or approximated by the rational number 119/59; in either case, worst rel. error would still be roughly 0.00078 .)
2) If our objective is to make the formula as accurate as possible for nearly circular ellipses, then it can be shown that p = 1/(2 ln(4/pi)) = 2.0698... should be used and that, when eccentricity e is small, relative error = (9 + 2/ln(pi/4))/16384 e^8 (1 + 2 e^2 +...). Of course, over all eccentricities, worst rel. error is not so good, now being roughly 0.0028 . But this value of p gives us another feature which can often be useful: We have an upper bound on the perimeter
4 (a + b) (pi/4)^((gm/am)^(1/(2 ln(4/pi)))) >= perimeter
with equality only when e = 0 or 1.
Best regards, David W. Cantrell

