 Euclidean similarity

A Euclidean similarity is a map $S:{\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfying $|S(x)-S(y)|=c|x-y|$ for a fixed $c>0$. It has the form $S(x)=cRx+b$ with $R$ orthogonal: after subtracting $S(0)$ and dividing by $c$, the map fixes zero and preserves distances, hence inner products by polarization, so it is a linear orthogonal map. It is a contraction if $c<1$. Contracting Euclidean similarities define self-similar compact sets; the open set condition is a useful separation hypothesis for computing their Hausdorff dimension.