Given a cuboid of dimensions $2 a \times 2 b \times 2 c$, and given a $3 \times 3$ rotation matrix $R$, I want to inscribe an ellipsoid whose axes are respectively along the directions specified by the columns of $R$. Find the lengths of the semi-axes along the three directions.
My thoughts:
The cuboid can be centered at the origin, and its faces made parallel to the coordinate planes. The equation of the ellipsoid then is
$ r^T Q r = 1 $
where $ r =[x,y,z]^T $ and $Q$ is a $3 \times 3$ symmetric positive definite matrix.
Tangency to the faces of the cuboid results in $3$ independent linear equations in the diagonal matrix of the reciprocal of the squared semi-axes lengths, which can be solved, and check for positiveness of all components of the solution vector.