when there is a projective transformation $\Phi$ such that $\Phi (v_i)=v_i'$. Notice, if two twisted $n$-gons are projectively equivalent, then their monodromies $M,M'$ are related by $M'=\Phi \circ M \circ \Phi^{-1}$.

The pentagram map on twisted $n$-gons commutes with projective transformations and as such descends to a map on the space ${\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ of projective equivalence classes of twisted $n$-gons.

We now recall a construction of coordinates on the space ${\mathcal{P}}_n \,/\, {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ of projective equivalence classes of twisted $n$-gons. These coordinates are known as *corner invariants* and were introduced in [@schwartz2008discrete].

Let $\{v_i \in {\mathbb P}^2\}$ be a twisted polygon. Then the corner invariants $x_i,y_i$ of the vertex $v_i$ are defined as follows. $$\begin{aligned}
\begin{split}
    x_i &:=\Big[
        v_{i-2},v_{i-1},\big((v_{i-2},v_{i-1})\cap (v_i,v_{i+1})\big), \big((v_{i-2},v_{i-1})\cap (v_{i+1},v_{i+2})\big)
        \Big], \\
    y_i &:= \Big[
        \big( (v_{i-2},v_{i-1})\cap (v_{i+1},v_{i+2})\big), \big((v_{i-1},v_i)\cap (v_{i+1},v_{i+2}) \big), v_{i+1}, v_{i+2}
    \Big],
\end{split} \label{eq2}\end{aligned}$$ where we define the cross-ratio $[a,b,c,d]$ of $4$ points $a,b,c,d$ on a projective line as $$\begin{aligned}
  [a,b,c,d]:=\frac{(a-b)(c-d)}{(a-c)(b-d)}. \label{eq1}\end{aligned}$$ Consider Figure \[Fig:CI\]. The value of $x_i$ is the cross ratio of the four points drawn on the line $(v_{i-2},v_{i-1})$ (i.e. the line on the left) and $y_i$ is the cross ratio of the four points drawn on the 