This is a basic example

$$\pi=\pi_{can}=\log|\cdot|:X=({\bf G}_{m}^{an})^{n}\to B_{0}=B={{\bf R}}^{n}$$

described in details in Appendix A. For any algebraic (or analytic) subvariety $Z\subset({\bf G}_{m}^{an})^{n}$ of dimension $m\leq n$ its image $\pi(Z)$ is a non-compact piecewise-linear closed subset of ${\bf R}^{n}$ of real dimension $m$. Smooth points for $\pi_{|Z}$ are dense in $\pi(Z)$.

In particular, if $Z$ is a curve then $\pi(Z)$ is a graph in $B$ with straight edges having rational directions. One can try to make a dictionary which translates the properties of the algebraic variety $Z\subset{\bf G}_{m}^{n}$ to the properties of the PL set $\pi(Z^{an})$ which is the closure of $\pi(Z(\overline{K}))$ in ${\bf R}^{n}$. This circle of ideas is a subject of the so-called “tropical geometry” (see e.g. [Mi]).