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

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.