```Date: Apr 27, 2009 8:22 AM
Author: David W. Cantrell
Subject: Re: A single term formula for calculating the circumference of ellipse

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/128-1921864-6683056-551607/circumference%20of%20an%20ellipse.pdf)Congratulations on finding a nice approximation! (Concerning your note, Ishould mention that your stated worst |rel. error| is overly pessimistic.Instead of saying |rel. error| < 0.0017, you should havesaid |rel. error| < 0.0013, but perhaps you simply made a typographicalmistake.)A few comments:In your formula, (pi/4) is raised to the power 4 a b/(a + b)^2. It isinteresting to note that that power can be written nicely in terms of thegeometric and arithmetic means of the semiaxes' lengths, namely, it is thesquare of the ratio of those means: Letting gm = sqrt(a b) andam = (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 bewritten as4 (a + b) (pi/4)^((gm/am)^2)But the fact that the ratio of means is squared is not crucial to theformula, and so it is natural to ask whether we could do "better" byraising gm/am to some power p other than 2.1)   If our objective is to minimize worst |rel. error|, then, by numericalmethods, it can be determined that p = 2.016861... should be used. We thenobtain |rel. error| < 0.00078 . (Of course, a disadvantage of usingp = 2.016861... is that it's not as easily remembered as 2. But p could, ifdesired, 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 fornearly 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, relativeerror = (9 + 2/ln(pi/4))/16384 e^8 (1 + 2 e^2 +...). Of course, over alleccentricities, 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 perimeter4 (a + b) (pi/4)^((gm/am)^(1/(2 ln(4/pi)))) >= perimeterwith equality only when e = 0 or 1.Best regards,David W. Cantrell
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