${{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ at the identity, and, upon a choice of basis, can be viewed as a $3 \times 3$ matrix defined up to addition of a scalar matrix. Our main result can thus be formulated as follows.

\[thm2\] The tangent vector to ${{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ represented by Glick’s operator $G_P$ coincides with the infinitesimal monodromy of $P$.

The proof will be given in Section \[Sec:proof\]. But first we need to check that Definition \[def:im\] makes sense, i.e. that the infinitesimal monodromy does not depend on the choice of the path $P_z$. This is established by the following:

Let $P_z$ and $\tilde P_z$ be two families of polygons such that $P_1 = \tilde P_1$ is a closed polygon and $\tilde P_z$ is projectively equivalent to $P_z$ for every $z$. Then, for the monodromies $M_z$ and $\tilde M_z$ of these families, at $z = 1$ we have $dM_z / dz = d\tilde M_z / dz$.

Let $\Phi_z$ be a projective transformation taking $P_z$ to $\tilde P_z$. Since $P_1 = \tilde P_1$, we have that $\Phi_1 = { {I}}$ (a generic $n$-gon in ${\mathbb P}^2$ does not admit any non-trivial projective automorphisms, provided that $n \geq 4$). Then we know that the 