Solve for ellipse axes given perimeter

I posted some notes this morning on how to find the perimeter of an ellipse given its axes. The notes include a simple approximation, a better but more complicated approximation, and the exact value. So given the semi axes a and b, the notes give three ways to compute the perimeter p.

If you are given the perimeter and one of the axes, you can solve for the other axis, though this involves a nonlinear equation with an elliptic integral. Not an insurmountable obstacle, but not trivial either.

However, the simple approximation for the perimeter is easy to invert. Since

$p \approx 2 \pi \left(\frac{a^{3/2} + b^{3/2}}{2} \right )^{2/3}$

we have

$a \approx \left( 2\left( \frac{p}{2\pi} \right)^{3/2} -\, b^{3/2}\right)^{2/3}$

The same equation holds if you reverse the roles of a and b.

If this solution is not accurate enough, it at least gives you a good starting point for solving the exact equation numerically.

If you’re not given either a or b, then you might as well assume a = b and so both equal p/2π.