A single term formula for calculating the circumference of ellipse

[Shahram Zafary]

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)

[Avni Pllana]

> My formula for calculating the circumference of
> ellipse is attached as a small note. I will be
> grateful if you read and judge it.

Hi Shahram,

here is a useful link, where you can compare your formula with the other known:

http://local.wasp.uwa.edu.au/~pbourke/geometry/ellipsecirc/

Best regards,
Avni

[Shahram Zafary]

Thank you my dear friend.
I read that before. I think my new formula about ellipse circumference compares very well with others. I posted it for others to know that really " I am right" or not.

[David W. Cantrell]

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

[Shahram Zafary]

Hello dear Mr. Cantrell.
Thanks a lot, for your really exact comments on my formula.
I am really grateful and hope you permit me contact you again about this formula
and some same findings that I found out before

[David W. Cantrell]

Dear Mr. Zafary,

You are certainly welcome to contact me using my sigmaxi.net email address.

I mentioned your nice formula to Gerard Michon recently, and he has now
added a section about it on his Numericana site. See

<www.numericana.com/answer/ellipse.htm#zafary>.

Note that Gerard found a very nice alternative way of writing your
approximation: The quantity h = ((a - b)/(a + b))^2 often appears in
dealing with the perimeter of an ellipse. Using h, your approximation can
be written simply as

pi (a + b) (4/pi)^h

Although I really liked your approximation before, I like it even more in
the form above!

Best regards,
David