determined by the segments $(p_i,p_{i+1})$, $p_i \in \mathcal{R}$. Then, the curve $C$ is oriented *counter-clockwise* with respect to the vector in $v$ normal to $T$. We select As the starting point for ordering $C$, we select the point $\tilde{p}$ such that: $$\tilde{p}=\operatorname*{argmax}\limits_{p_i\in\mathcal{R}} h(p_i).$$ In case of symmetries around a vertex, multiple choices of the starting point are possible: we select the candidate point that is the farthest from the other elements of $\mathcal{R}$. The stability of the starting point of a $ring$ is confirmed in numerous experiments we performed on meshes of different resolution, where by mesh resolution we mean the number of vertices of the mesh. Figure \[fig:ring\_stab\] shows the vector field generated by the difference between $\tilde{p}$ and $v$ ($\overrightarrow{{\tilde{p}-v}}$) all over the mesh. The orientation of the field indicates the position of $\tilde{p}$. The pictures show a detail of the field over a mesh with $40K$ vertices and two mesh sub-samplings with $16K$ and $8K$ vertices: the overall orientation of the field (and therefore the choice of $\tilde{p}$) is robust to different mesh samplings.

![Arrows represent the starting point of the rings in meshes representing the same surface but sampled with a different number of vertices (40K, 