Modifying Ramanujan's second approximation for the perimeter of an ellipse
David W. Cantrell
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
ellipse, the more accurate of which is often written as
[please view in a fixed-width font]
3h
pi(a + b) ( 1 + ----------------- )
10 + Sqrt(4 - 3h)
where a and b are the lengths of the semiaxes of the ellipse and
h = ((a-b)/(a+b))^2. This approximation is of truly remarkable accuracy for
ellipses of low eccentricity; and its worst relative error, occurring when
eccentricity e = 1, is 7/22 pi -1, or about -4*10^(-4).
A natural question to ask is whether this approximation can be improved in
some attractive way. One possible way would be to add a corrective term c,
so that the approximation would look like
pi(a + b) (1 + (original fraction) + c).
Indeed, just yesterday I received an email from one Edgar Erives specifying
an expression -- unfortunately, rather complicated -- for such a corrective
term. However, there is a simple corrective term which may be used:
c = (4/pi - 14/11) h^12
which I designed to precisely correct the approximation when e = 1 and
reduce worst |relative error|, while being benign with respect to the
accuracy of the approximation when e is small.
We may then approximate the perimeter of an ellipse,
with |relative error| < 1.5*10^(-5), by
3h 12
pi(a + b) (1 + ----------------- + (4/pi - 14/11) h ).
10 + Sqrt(4 - 3h)
[By the way, the exponent 12 is not "precise", in the sense that one could
numerically determine a nonintegral exponent which would give a slightly
smaller max|relative error|. Nonetheless, it would still exceed
1.4*10^(-5), and so, in the interest of computational efficiency, it seems
nicest to use exactly 12 for the exponent.]
Those who would like to have a simple approximation providing even smaller
worst |relative error| may be interested in item (2) at the end of my
recent article "Two New Approximations, in a Certain Form, for the
Perimeter of an Ellipse" at
<http://mathforum.org/discuss/sci.math/m/604793/604793>. However, the
accuracy of (2) for near-circular ellipses is far worse than that of the
approximation presented here.
Has anyone seen this "corrected" version of Ramanujan's approximation
before?
David W. Cantrell