What is the geometric interpretation of the Arithmetic–geometric mean?

The arithmetic–geometric mean of 2 values $a_0$,$b_0$, is the value to which the arithmetic and geometric values converge, being $$a_n=\frac{a_{n-1}+b_{n-1}}{2} \text{ and } b_n=\sqrt{a_{n-1} .b_{n-1}}$$ with $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = AGM(a_0,b_0)$$

The arithmetic and the geometric values have a simple geometrical interpretation

if $a_0$ is the segment FE, and $b_0$ is the segment EG, then the radius CD is the arithmetic mean, and the segment ED is the geometric mean.

Is possible to draw the arithmetic–geometric mean on that circle (or there is another geometrical interpretation maybe not involving circles)?

Solution 1:

The angular average of the distance between the perimeter of an ellipse and its centre:

\begin{align*} \langle r \rangle_{\theta} &= \frac{1}{2\pi} \int_{0}^{2\pi} r \, d\theta \\ &= \frac{1}{2\pi} \int_{0}^{2\pi} \frac{a b}{\sqrt{a^2\sin^2 \theta+b^2\cos^2 \theta}} \, d\theta \\ &= \frac{2ab}{\pi} \int_{0}^{\pi/2} \frac{d\theta}{\sqrt{a^2\sin^2 \theta+b^2\cos^2 \theta}} \\ &= \frac{ab}{\operatorname{agm}(a,b)} \end{align*}