  We use the Schwarz-Christoffel formula to show that for every $n\geq 3$, the
space of labelled immersed $n$-gons in the plane up to similarity is
homeomorphic to $\mathbb{R}^{2n-4}$. We then prove that all immersed triangles,
quadrilaterals, and pentagons are embedded, from which it follows that the
space of labelled simple $n$-gons up to similarity is homeomorphic to
$\mathbb{R}^{2n-4}$ if $n\in \{3,4,5\}$. This was first shown by Gonz\'ales and
L\'opez-L\'opez for $n=4$ and conjectured to be true for every $n\geq 5$ by
Gonz\'alez and Sedano-Mendoza.
