 Affine image of a circle

Let a radius-$R$ circle have center $c\in {\mathbb{R}}^2$, and apply the real affine map $y=Mx+b$. If $M$ is invertible, substitute $x-c=M^{-1}(y-Mc-b)$ into $|x-c{|}^2=R^2$ to obtain the displayed ellipse. Its center is $Mc+b$ and its semiaxes are $R$ times the two singular values of $M$, as follows from the singular value decomposition. A rotation or reflection preserves both semiaxes and hence gives another circle. If $M$ has rank one, the image is a segment; rank zero gives one point. This real-plane description also handles maps that stretch real and imaginary coordinates differently and are not holomorphic complex maps.