Approximating the Circumference of an Ellipse

The realization that the circumference of a circle is related in a simple way to the diameter came at an early stage in the development of mathematics. But who was first to prove that all circles are similar, with the ratio of circumference {C} to diameter {D} the same for all? Searching in Euclid’s Elements, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that {C / D} is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post here.

No Simple Formula for the Ellipse

While the elementary formula {C = 2\pi r} holds for all circles, there is no corresponding simple formula for the ellipse. The ellipse has two length scales, the semi-major axis {a} and the semi-minor axis {b} but, while the area is given by {\pi a b}, we have no simple formula for the circumference.

We can integrate the element of arc-length around the ellipse to obtain an expression for the circumference:

\displaystyle C = 4 a \int_ 0^{\pi/2} \sqrt {1 - e^2\sin^2\theta} \,\mathrm{d}\theta \ \ \ \ \ (1)

The limiting values {2\pi a} for {e=0} and {4a} for {e=1} are immediate but, in general, there is no simple analytical expression for the integral. The integral in (1) is called the complete elliptic integral of the second kind, which is not an elementary function. We can expand the integrand and integrate term by term to get an infinite series. But it is of interest to seek more simple expressions for the circumference.

Averaging the Axis Lengths

For the circle we have one radius {r}; for the ellipse we have two semi-axes {a} and {b}. It makes sense to average {a} and {b} in some way and use the average in the formula {C=2\pi\bar r}. In the Figure below we plot the circumference as a function of eccentricity, using {C=2\pi \bar r} with arithmetic mean {\bar r=(a+b)/2} (left), geometric mean {\bar r=\sqrt{ab}} (centre) and root mean square value {\bar r=\sqrt{(a^2+b^2)/2}} (right). All approximations are accurate for small {e}, but all deviate from the true value as {e} approaches {1}.

Circumference length as a function of eccentricity, using {C=2\pi \bar r} with {\bar r=(a+b)/2} (left), {\bar r=\sqrt{ab}} (centre) and {\bar r=\sqrt{(a^2+b^2)/2}} (right). Thick line is exact value.

Can we combine these results to obtain a better formula? Yes! The exact value for {e=1} is {C=4}. At this limit, the arithmetic mean (AM) is {\bar r=(a+b)/2 = a/2} so {C=\pi}, which is too small. The root mean square (RMS) is {\bar r=\sqrt{(a^2+b^2)/2} = a/\sqrt{2}}, so {C= \sqrt{2}\pi}, which is too large. Taking a weighted linear combination {\alpha\mathrm{(AM)}+ \beta\mathrm{(RMS)}} and requiring this to be exactly correct at {e=0} and {e=1}, we find that {\alpha = (\sqrt{2}-4/\pi)/(\sqrt{2}-1)} and {\beta = (4/\pi-1)/(\sqrt{2}-1) }, so that {\alpha \approx 0.3403} and {\beta \approx 0.6597}. To simplify the formula we take {\alpha=1/3} and {\beta=2/3} and write

\displaystyle C = 2\pi \left[ {\textstyle{\frac{1}{3}}}\mathrm{(AM)} + {\textstyle{\frac{2}{3}\mathrm{(RMS)}}} \right] = \frac{2\pi}{6} \left[ (a+b) +2\sqrt{2(a^2+b^2)} \right] \ \ \ \ \ (2)

The graph of formula (2), together with graph of its error, is seen in the Figure below. The maximum error is everywhere less than 2%, which is good enough for many applications.

Left: Circumference from AM, RMS and (2). Right: Percentage error of (2).

Ramanujan’s Wonderful Formulae

For astronomical purposes it is often vital to have high accuracy. The Indian genius Srinivasa Ramanujan discovered a number of remarkable formulas that give the circumference to excellent precision. The first of these formulas is

\displaystyle C = \pi \left[3 (a + b) - \sqrt {(3 a + b) (a + 3 b)} \right] \ \ \ \ \ (3)

The error in this formula is less than {0.4\%} throughout the full range of eccentricity. An even more precise formula of Ramanujan is given by

\displaystyle C = \pi (a + b) \left[ 1 + \frac {3 h} {10 + \sqrt {4 - 3 h}} \right] \ \ \ \ \ (4)

where {h = (a - b)^2/(a + b)^2 }. This has an error less than {0.04\%} throughout the full range. The values of the circumference as a function of eccentricity for Ramanujan’s two formulas are plotted in the Figure below. The thick black line is the exact value, and the thinner coloured lines are for the approximate formulas. It is not possible to distinguish by eye any difference between the approximate and the exact results.

Exact circumference (thick black line) and Ramanujan’s approximations (3) (coloured, left) and (4) (coloured, right).

Many other approximate formulas have been found, from the time of Kepler up to the present day. For more information, see the “Ellipse” article on Wikipedia. Matt Parker has a video in his “Stand Up Maths” series about this topic.

Sources

{\bullet} Matt Parker: Why is there no equation for the perimeter of an ellipse? Stand Up Maths video on YouTube Here.

{\bullet} Wikipedia article Ellipse: \url{http://www.wikipedia.org/}

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