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 $d_1$, $d_2$, $d_{\mathrm{\infty }}$ 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 $\mid A\mid$ 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.