# Affine transformation mapping circle to ellipse

I'm trying to find an affine transformation that maps the unit circle to an ellipse centered at $$(1,3)$$ such that points P$$(-3,-1)$$ and R$$(5,7)$$ are at the greatest distance from the centre of the ellipse along the major axis while points Q$$(0,4)$$ and S$$(2,2)$$ are at the greatest distance from the centre of the ellipse along the minor axis.

I think we need to rotate the unit disc followed by scaling and then another rotation. But I don't know how to find the matrix that does that. Ultimately, I think there will be translation by $$(1,3)^T.$$

Any help appreciated.

• $$(5,7), (1,3), (-3,-1)$$ lie on line $$y=mx+c$$
• $$(0,4), (1,3), (2,2)$$ lie on line $$y=\ldots$$
• Conclude this is rotation with $$\theta = \tan^{-1} m$$
• Use rotation matrix $$\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}$$
• Find lengths of axes of ellipse. Hence find scaling matrix. $$\begin{bmatrix} a & 0 \\ 0 & b \\ \end{bmatrix}$$
• So, am I right in saying that we have to rotate the unit disc by the rotation matrix where $\theta = \dfrac{\pi}{4}$ followed by scaling matrix where $a$ and $b$ are $4$ and $1$ respectively and then translation by $(1,3)^T$ ? So do we only need one rotation ?
• @Rhombus Almost! It's $4\sqrt{2}$, $\sqrt{2}$. Commented Nov 17, 2020 at 15:33