Polygonal arcs and polygons as non-self-intersecting finite unions of line segments in $\mathbb R^2$

Work in the metric plane $\mathbb R^2$ of $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it. A polygonal arc from $x$ to $y$ is the image of an injective continuous map $\gamma:[0,1]\to\mathbb R^2$ (Injection, surjection, bijection) for which there are finitely many parameters $0=t_0<\cdots<t_m=1$ such that $\gamma$ is affine and nonconstant on each $[t_{i-1},t_i]$. Its vertices are the finitely many points $\gamma(t_i)$ (The cardinality $\lvert A\rvert$ of a finite set). This is a simple polygonal path in the terminology of Polygonal paths and polygonally connected subsets of $\mathbb{R}^n$.

A polygon is the image of a continuous map $\gamma:[0,1]\to\mathbb R^2$ with $\gamma(0)=\gamma(1)$, affine and nonconstant on finitely many consecutive parameter intervals, and injective on $[0,1)$. Nonconsecutive constituent segments are disjoint, and consecutive ones meet only at their common endpoint. A polygon is therefore a simple closed polygonal curve, not the filled region it may bound.