The question was inspired by this answer.
Condition for perfect packing of ellipse with circles: $n$ circles can be perfectly packed along the major axis of the ellipse with semi-axes $a,b\quad$ iif
\begin{align} \frac ba&=\sin\frac{\pi}{2\,n} \tag{1}\label{1} . \end{align}
Here "perfectly packed" means that the chain of circles with the centers on the major axis is inscribed into the ellipse, and the radii of the circles at both ends agree with the curvature of the ellipse at the ends of the major axis.
Condition \eqref{1} works for both odd (ex. $n=7$)
and even $n$ (ex. $n=8$)
Question:
Is this a well-known condition? Any reference?
Addendum
General formula for the radius of $k$-th circle in a chain of $n$ perfectly packed circles along the major axis of the ellipse \begin{align} r_k&=a\,\sin\frac{\pi}{2\,n}\,\sin\frac{(2\,k+1)\,\pi}{2\,n} ,\quad k=0,\dots,n-1 , \end{align} where $a$ is the major semi-axis, and the location of the center of $k$-th circle is found as \begin{align} O_k&= F_1\cdot\cos^2 \frac{(2\,k+1)\,\pi}{4\,n} +F_2\cdot\sin^2 \frac{(2\,k+1)\,\pi}{4\,n} , \end{align} where $F_1,F_2$ are the focal points of the ellipse.
Distances from $F_1,F_2$ to the tangent point of the $k$-th circle with the ellipse are
\begin{align} |F_1T_k| &= 2\,a\,\sin^2\frac{(2\,k+1)\,\pi}{4\,n} ,\\ |F_2T_k| &= 2\,a\,\cos^2\frac{(2\,k+1)\,\pi}{4\,n} . \end{align}
\begin{align} \cos\angle T_kO_kF_2&= \frac{\sin\frac{k\,\pi}n-\sin\frac{(k+1)\,\pi}n} {\sin\frac{k\,\pi}n+\sin\frac{(k+1)\,\pi}n} . \end{align}
