Stereographic compactification and affine bi-Lipschitz homeomorphisms

Let ${\sigma }_q:{\mathbb{R}}^q\to {\mathbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with centre the north pole $N_q$. Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$, for $i=1,2$. Let $\mathrm{\Phi }:W_1\to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism ${\sigma }_{q_2}\circ \mathrm{\Phi }\circ {\sigma }_{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ 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.