Affine transformation mapping circle to ellipse

Question

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.

Answer 1

Hints :

• $(5,7), (1,3), (-3,-1)$ 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. $$ \begin{bmatrix} a & 0 \\ 0 & b \\ \end{bmatrix} $$