Rapidly convergent series for ellipse perimeter

The Endeavour 2024-06-20

The previous post looked at two simple approximations for the perimeter of an ellipse. Approximations are necessary since the perimeter of an ellipse cannot be expressed as an elementary function of the axes.

Kepler noted in 1609 that you could approximate the perimeter of an ellipse as the perimeter of a circle whose radius is the mean of the semi-axes of the ellipse, where the mean could be either the arithmetic mean or the geometric mean. The previous post showed that the arithmetic mean is more accurate, and that it under-estimates the perimeter. This post will explain both of these facts.

There are several series for calculating the perimeter of an ellipse. In 1798 James Ivory came up with a series that converges more rapidly than previously discovered series. Ivory’s series is

$P = \pi (a+b) \left(1 + \sum_{n=1}^\infty \left(\frac{(2n-1)!!}{2^n n!}\right)^2 \frac{h^n}{(2n-1)^2}\right)$

where

$h = \frac{(a-b)^2}{(a+b)^2}$

If you’re not familiar with the !! notation, see this post on multifactorials.

The 0th order approximation using Ivory’s series, dropping all the infinite series terms, corresponds to Kepler’s approximation using the arithmetic mean of the semi-axes a and b. By convention the semi-major axis is labeled a and the semi-minor axis b, but the distinction is unnecessary here since Ivory’s series is symmetric in a and b.

Note that h ≥ 0 and h = 0 only if the ellipse is a circle. So the terms in the series are positive, which explains why Kepler’s approximation under-estimates the perimeter.

Using just one term in Ivory’s series gives a very good approximation

$h = \pi(a + b)\left(1 + \frac{1}{4} \frac{(a-b)^2}{(a+b)^2}\right)$

The approximation error increases as the ratio of a to b increases, but even for a highly eccentric ellipse like the orbit of the Mars Orbital Mission, the ratio of a to b is only 2, and the relative approximation error is about 1/500, about 12 times more accurate than Kepler’s approximation.

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