# Analytic Form Fitting in Poor Triangular Meshes
## 3 Methodology
### Extraction of Subset \(S\) Based on Dibedral Angle  
This method for extracting from \(M\) the triangle subset \(S\) (\(\subset M\)) which contains the seed triangle \(t_{s}\), is applicable when the EDGE border between \(S\) and \(M-S\) holds only \(C^{0}\)-continuity (i.e., \(\partial S\) is a sharp border). In this case, (a) \(S\) is indifferent to the \(\widehat{S}\) type of analytic form sought and (b) the extraction of \(S\) precedes and is independent from the \(g()\) fitting process.  
In this manuscript, we say that two EDGE-sharing triangles \(t_{i},t_{j}\in M\) hold only \(C^{0}\)-continuity if their dihedral angle is above a threshold \(a_{max}\). The dihedral angle \(\angle(t_{i},t_{j})\)  
Figure 11: Cone 2D example. Vectors \(e_{i}\) point to \(p_{v}\) and are perpendicular to \(n_{i}\).  
between triangles \(t_{i}\) and \(t_{j}\) is the one between their normal vectors (under a uniform LOOP traversal direction). We measure the dihedral angle in the interval \([0,180)\) degrees. Notice that \(\angle(n_{1},n_{2})=180\) implies that \(M\) would be non-manifold.  
The first method for extraction of the triangle subset \(S\) is based on the dihedral angle between the triangles of \(M\). This method is employed when the target segment of the triangulation holds strict \(C^{0}\)-continuity with respect to the other segments of \(M\). This extraction process is indifferent to the type \(\widetilde{S}\) of the analytic form and occurs separately from the fitting process as shown in Figure 2 (red).  
**Input:**  
1. \(M(\mathcal{T},\mathbf{P})\): triangle (two-manifold) set with geometry \(\mathbf{P}\) and topology \(\mathcal{T}\).
2. \(t_{s}\): selected seed triangle from \(\mathcal{T}\).
3. \(\alpha_{max}\): maximum angle value for dihedral angle-based extraction.
4. \(\mathbf{E}_{\alpha}()\): dihedral angle equivalence relation defined on \(M\).  
**Output:**  
1. \(S\): the equivalence class (or block) containing \(t_{s}\), of \(M\) under the equivalence relation \(\mathbf{E}_{\alpha}()\).  
**Definition. Equivalence Relation \(\mathbf{E_{\alpha}()}\)**  
The triangles are equivalent under \(\mathbf{E}_{\alpha}()\) if there is a triangle sequence (Figure 12) \(L=[t_{0},t_{1},...,t_{f}]\) such as:  
1. \(L\subset M\)
2. Any two consecutive triangles of \(L\), \(t_{i}\) and \(t_{i+1}\) share an EDGE.
3. \(\angle(t_{i},t_{i+1})\ <\alpha_{max}\)  
The equivalence character of relation \(\mathbf{E}_{\alpha}()\) is not proved here. However, it is easy to verify that \(\mathbf{E}_{\alpha}()\) is symmetric, reflexive and transitive. The reason we show this small digression here is that the \(g()\) support subset \(S\subset M\) containing the seed triangle \(t_{s}\) is computable (when \(S\) and \(M-S\) hold only \(C^{0}\)-continuity at \(\partial S\)) by using well known algorithms for transitive closure.  
The boundary representation of input triangle set \(M\) supports FACE, VERTEX, EDGE neighborhood interrogations in constant time. Therefore, the dihedral angle-based extraction process of the \(g()\) support subset \(S\) presents low computational cost and fast execution times.