When resolving singularities in toric surfaces, one looks for some subdivision $\Delta'$ of the defining fan $\Delta$ \Delta \subset \mathbb{R}^2$of the variety (such surface. Such a subdivision corresponds to a proper birational map$X(\Delta')\to X(\Delta)$, giving the resolution of singularities. The 1-dimensional rays of this subdivision is found using the Hirzebruch-Jung continued fractions. As Alexey remarks, the convergents of this continued fraction gives the vertices of the convex hull of$(\Delta\cap \mathbb{Z}^2)\setminus {(0,0)}$. The above image is taken from Jon Voigt's paper 1 When resolving singularities in toric surfaces, one looks for some subdivision$\Delta'$of the defining fan$\Delta$of the variety (such a subdivision corresponds to a proper birational map$X(\Delta')\to X(\Delta)\$. The 1-dimensional rays of this subdivision is found using the Hirzebruch-Jung continued fractions.