Bi-Lipschitz map ($m{d}_X(x,y)\le d_Y(f(x),f(y))\le M{d}_X(x,y)$)

A bi-Lipschitz map between metric spaces satisfies the displayed bounds for all input pairs, with constants $0<m\le M<\infty$. 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.