= Affine image of a circle
{title2=$(y-Mc-b)^T(MM^T)^{-1}(y-Mc-b)=R^2$}
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.
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