= Bi-Lipschitz map
{title2=$m\,d_X(x,y)\leq d_Y(f(x),f(y))\leq M\,d_X(x,y)$}
A bi-Lipschitz map between <metric spaces> satisfies the displayed bounds for all input pairs, with constants $0<m\leq M<\infty$. The lower bound makes it <injective>; its inverse on its image is <Lipschitz continuous> with bound $1/m$, while the map itself has <Lipschitz bound> $M$. Thus it controls both stretching and compression. It is a <homeomorphism> onto its image, preserves <Hausdorff dimension>, and has <bi-Lipschitz distortion> at most $M/m$. Uniformly multiplying every distance by the same positive factor has distortion one, as explained in <metric embedding>.
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