  In this paper we study the space $\mathbb{L}(n)$ of $n$-gons in the plane
degenerated to segments. We prove that this space is a smooth real submanifold
of $\mathbb{C}^n$, and describe its topology in terms of the manifold
$\mathbb{M}(n)$ of $n$-gons degenerated to segments and with the first vertex
at 0. We show that $\mathbb{M}(n)$ and $\mathbb{L}(n)$ contain straight lines
that form a basis of directions in each one of their tangent spaces, and we
compute the geodesic equations in these manifolds. Finally, the quotient of
$\mathbb{L}(n)$ by the diagonal action of the affine complex group and the
re-enumeration of the vertices is described.
