Given two ellipses, does there exist an explicit transformation between them?

Question

Say you have the standard parameters (axes lenghts, angle of rotation) of two different ellipses. Is there a swift way of transforming the points of one ellipse to the points of the other?

Thanks!

Answer 1

Yes. Any ellipse with $x$-radius $a$ and $y$-radius $b$ is obtained from the unit circle by expanding the $x$-axis by a factor of $a$ and the $y$ axis by a factor of $b$. If the ellipse is rotated, one has to use the usual rotation matrix $$(x,y)\mapsto \begin{pmatrix}\cos\theta &\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}$$If you want the centers to move, then use $x\mapsto x-k$ and $y\mapsto y-h$ appropriately.

Answer 2

Find an affine (i.e. linear+constant) transformation that takes the centre of one ellipse to the centre of the other and the major and minor axes of one to the major and minor axes of the other. Then this transforms the first ellipse onto the other.