closed curves defined by the set of points $P_i \in \mathcal{R}$; (b): the black dots correspond to the elements $p_i$ of the three central curves in (a).[]{data-label="fig:ring_construction"}](pic/ring_pi.png "fig:"){width="5.5cm"}
                                                                                                                                   (a)                                                                                                                                                                                                                                                                  (b)
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Such an intersection is represented by the set of points $\mathcal{R}=\{p_1, p_2, \ldots, p_k\}$ that approximate the intersection between the sphere and the surface. Figure \[fig:ring\_construction\] shows a number of concentric rings over a triangle mesh. To determine a $ring$ around a vertex $v$, we follow a mesh expansion approach driven by the Euclidean distance from the vertex $v$, as summarized in the following steps:

1.  All the edges that are incident in $v$ are added to a list $L$.

2.  Starting from an edge $e=(v,v_1) \in L$, the intersection between $e$ and the sphere centered in $v$ with radius $R$ is evaluated. If there actually is an intersection, it is stored as a new point $p_i$, otherwise, if $e$ completely falls inside the sphere, we add to $L$ all the edges that are incident to $v_1$. The edge $e$ is removed from $L$ and labeled as visited.The value $h(p_i)$ on $p_i$ is given by the linear interpolation of the values that $h$ assumes in $v$ and $v_1$.

