From: David W. Cantrell <DWCantr...@sigmaxi.org>
Date: 24 May 2004 18:41:14 GMT
Local: Mon, May 24 2004 2:41 pm
Subject: Modifying Ramanujan's second approximation for the perimeter of an ellipse
In this short article, a simple modification of the more accurate of
Ramanujan's well known approximations for the ellipse perimeter is
presented. The modification is notable because (1) it substantially reduces
the worst |relative error|, (2) it does so at low computational cost, and
(3) it does not disturb the original approximation's high accuracy when
eccentricity is small.
In 1914, Ramanujan presented two approximations for the perimeter of an
where a and b are the lengths of the semiaxes of the ellipse and
A natural question to ask is whether this approximation can be improved in
pi(a + b) (1 + (original fraction) + c).
Indeed, just yesterday I received an email from one Edgar Erives specifying
c = (4/pi - 14/11) h^12
which I designed to precisely correct the approximation when e = 1 and
We may then approximate the perimeter of an ellipse,
[By the way, the exponent 12 is not "precise", in the sense that one could
Those who would like to have a simple approximation providing even smaller
Has anyone seen this "corrected" version of Ramanujan's approximation
David W. Cantrell
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