In 1609, Kepler remarked that the perimeter of an ellipse with semiaxes *a* and *b* could be approximated either as

*P* ≈ 2π(ab)^{½}

or

*P* ≈ π(*a* + *b*).

In other words, you can approximate the perimeter of an ellipse by the circumference of a circle of radius *r* where *r* is either the geometric mean or arithmetic mean of the semi-major and semi-minor axes.

How good are these approximations, particularly when *a* and *b* are roughly equal? Which one is better?

When can choose our unit of measurement so that the semi-minor axis *b* equals 1, then plot the error in the two approximations as *a* increases.

We see from this plot that both approximations give lower bounds, and that arithmetic mean is more accurate than geometric mean.

Incidentally, if we used the geometric mean of the semi-axes as the radius of a circle when approximating the *area* then the results would be exactly correct. But for perimeter, the arithmetic mean is better.

Next, if we just consider ellipses in which the semi-major axis is no more than twice as long as the semi-minor axis, the arithmetic approximation is within 2% of the exact value and the geometric approximation is within 8%. Both approximations are good when *a* ≈ *b*.

The next post goes into more mathematical detail, explaining why Kepler’s approximation behaves as it does and giving ways to improve on it.

Nice!

This seems a case for : P ≈ π(a + b) is more clearly about the arithmetic mean if written as P ≈ 2π(a + b)/2. And that also better brings out the symmetry between these two approximations.

Note that several time (mostly in images), the _harmonic_ mean is mentioned?

back in trhe early 60’s in engineering drawing, we learned the four center ellipse (drawing) approximation I wonder how the lentgh of this approximation compares to the correct value Too lazy to figure it out right now maybe later