in ${\mathbb P}^2$. Let $[P] \in {\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ be its projective equivalence class. Then, applying the scaling transformation $R_z$ given by to $[P]$, we get a path $R_z[P]$ in $ {\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ such that $R_1[P] = [P]$. Now, choose a smooth in $z$ lift $P_z$ of the path $R_z[P]$ to the space ${\mathcal{P}}_n$ of actual twisted polygons such that $P_1 = P$ (we will construct an explicit example of such a lift later on). Denote by $M_z \in {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ the monodromy of $P_z$. It is a family of projective transformations such that $M_1$ is the identity, $M_1 = { {I}}$. This family [does]{} depend on the choice of the lift $P_z$ of the path $R_z[P]$. However, as we show below, the tangent vector $dM_z/dz$ at $z = 1$ does not depend on that choice, and this is what we call the *infinitesimal monodromy*.

\[def:im\] The *infinitesimal monodromy* of a closed polygon $P$ is the derivative $dM_z / dz$ at $z=1$, where $M_z$ is the monodromy of any path $P_z$ of polygons such that $P_1 = 1$ and $[P_z] = R_z[P]$.

The infinitesimal monodromy is therefore a tangent vector to the projective group 