On the one dimensional polynomial and regular images of ${\mathbb{R}}^n$

In this work we present a full geometric characterization of the 1-dimensional polynomial and regular images $S$ of ${\mathbb{R}}^n$ and we compute for all of them the invariants $\mathrm{p}(\mathrm{S})$ and $\mathrm{r}(S)$, already introduced in \cite{fg2}.