Let σq : Rq → Sq \ Nq be the inverse of the stereographic projection with centre the north pole Nq. Let Wi be a closed subset of Rqi , for i = 1; 2. Let Φ : W1 → W2 be a bi-Lipschitz homeomor-phism. The main result states that the homeomorphism σq2 οΦοσ-1q1 is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at Nq1 with value Nq2 whenever W1 is unbounded.
As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: 1) Sampaio's tangent cone result; 2) Links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette.