## 3 Parametric Model Definition
### Surface Meshing Method
#### 3.1.3. Compass Method Overview  
The Compass Method (CM), also known as Tchebyshev net [2; 19], is a well-described method for surface meshing in Parametrized Surface Theory [33], corresponding to a mesh over a parametrized surface S: \(\mathrm{U}\subset\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}\), whose directional derivatives (\(S_{\mathrm{u}}\) and \(S_{\mathrm{v}}\)  
Figure 5: Generation of a geodesic curve from a point and a direction vector: (**a**) First iteration from the origin point; (**b**) Second iteration from the point resulting from the first iteration; (**c**) Resulting geodesic curve. Figure 6: Geodesic master curves from a starting point and two directions. over the surface coordinate field satisfy the requirement that the partial derivatives of one parameter have a constant length along the parameter lines of the other parameter:  
\[\left|S_{\mathrm{u}}\right|_{\mathrm{v}}=\left|S_{\mathrm{v}}\right|_{\mathrm{u}} \tag{5}\]  
The CM has a geometrical approach, shown in Figure 7, where the surface meshing is the result of the intersection of an array of spheres with a given radius all over a surface. Starting from a pair of intersecting curves on the surface, the curves are subdivided by placing spheres with the given radius and using the last intersection obtained recursively as the centre point (Figure 7a). The resulting intersection points on each curve serve as centre points for a tuple of iterating spheres whose intersections with their own and with the surface generate a new array of centre points for the subsequent iterations (Figure 7b). Finally, once the boundary is reached, a two-dimensional net of points is obtained, which can be used as constitutive points for a network of two-dimensional interpolated curves over the surface (Figure 7c). It is a known problem that meshing surfaces with high local curvature lead to accuracy errors when applying the CM. As mentioned above, the CM computes each point on the mesh by intersecting spheres with the base surface. The desired length for the subdivision over the surface would be the radius of the spheres. To illustrate this problem, Figure 8 shows the analogous issue in planar geometry. For the sake of simplicity, given a circular curve \(C\) and a starting point \(O_{0}\) on the curve, a circumference with radius \(r\) can be traced giving the intersection point \(O_{1}\) on \(C\). The distance between \(O_{0}\) and \(O_{1}\) is \(r\) and corresponds to the chord of the segment of \(C\) between \(O_{0}\) and \(O_{1}\). At the same time, finding the endpoint \(O_{1^{\prime}}\) of a segment of \(C\) with length \(r\) starting from \(O_{0}\), it is observed that \(O_{1}\) and \(O_{1^{\prime}}\) are not coincident, and the distance between them corresponds to the CM error, which depends on the average curvature of each segment of \(C\). Only if \(C\) were a straight line would \(O_{1}\) and \(O_{1^{\prime}}\) coincide. For the circular curve \(C\) with radius \(R\) in Figure 8, the error due to its curvature can be easily calculated as the difference between the arch length of \(C\) for the angle \(\alpha\) between  
Figure 8: Compass Method accuracy error. Figure 7: Compass Method: (**a**) Initial curves subdivision; (**b**) Surface subdivision process; (**c**) Resulting mesh. and \(O_{1}\), and the arch length of \(C\) between \(O_{0}\) and \(O_{1^{\prime}}\) (segment of \(C\) with length \(r\)), which has the same length as the chord of \(C\) for the same opening angle \(\alpha\):  
\[\text{Arch length}_{O_{0}-O_{1}}=R\;\alpha, \tag{6}\]  
\[\text{Arch length}_{O_{0}-O_{1}^{\prime}}=\text{Chord}_{\alpha}=2R\sin(\alpha/2), \tag{7}\]  
\[\text{Error}_{\alpha}=\text{Arch length}_{\alpha}-\text{ Chord}_{\alpha} \tag{8}\]  
The potential impact of this error from applying the CM on a gridshell depends on the length of the laths, the intended mesh subdivision distance, and surface extent, but mainly on the average curvature between each pair of consecutive points.