Let $f(z,w)=zw+z+w-1$. Then ${V}^{\circ}\subset(\mathbb{C}^{*})^{2}$ is a hyperbola. The Newton polygon $\Delta$ is a square $\{(x,y)\in\mathbb{R}^{2}\ |\ 0\leq x\leq 1,0\leq y\leq 1\}$ and the corresponding toric surface $\mathbb{C}T_{\Delta}$ is the hyperboloid ${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$.

Take now $\bar{\Delta}=\{(x,y)\in\mathbb{R}^{2}\ |\ 0\leq x,0\leq y,x+y\leq 1\}$. The corresponding toric surface is ${\mathbb{C}}{\mathbb{P}}^{2}\supset(\mathbb{C}^{*})^{2}$. The images of ${V}^{\circ}$ under the associated moment maps are sketched on Figure 4.