= Euclidean similarity
{c}
{title2=$S(x)=cRx+b$}
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>.
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