## 3 Parametric Model Definition
### Surface Meshing Method
#### 3.1.3. Compass Method Overview
Figure 9 shows the total deviation in millimetres for five different curve lengths, varying the curvature radius between 5 m and 15 m and considering a usual subdivision radius of 1 m. It is observed that for a 30 m long lath with 5 m radius of curvature, a total error of 50.3 mm is obtained due to the accumulative effect of the described CM accuracy error. The total error decreases exponentially as the radius of curvature increases. To overcome this drawback, the Improved Compass Method (ICM) was developed [20]. It mitigates such accuracy error through the following procedure: having defined the desired grid node distance \(r\) and a resolution factor \(S\), the algorithm runs the CM using a subdistance \(r/S\) and dismissing the intermediate \(S\)-1 subdivision points before generating the mesh curves. Again, to illustrate the method in Figure 10, the ICM is applied on a planar curve and compared with the CM to show the improvement in accuracy. Given a curve \(C\) and a starting point \(O_{0}\), applying a CM single step with a desired subdivision distance \(r\), the point \(O_{1}\) over the curve is obtained. To apply the ICM over \(C\) with a subdivision distance \(r\), first, the resolution factor \(S\) must be defined, in this case, 2. Given \(S=2\), ICM uses \(r/2\) as the radius for the intersecting circles and has to perform 2 steps to obtain the subdivision point analogous to the CM subdivision point \(O_{1}\). The first circle with a centre at \(O_{0}\) and radius \(r/2\) gets the intersection point \(O_{1^{\prime}}\) over C. The second step uses \(O_{1^{\prime}}\) as the centre for the second intersecting circle with the same radius \(r/2\), obtaining \(O_{2^{\prime}}\) over \(C\). Finally, the ICM removes \(S\)-1 intermediate subdivision points, in this case, 1 intermediate subdivision point (\(O_{1^{\prime}}\)), keeping \(O_{2^{\prime}}\) as the final point. The distance between the CM's \(O_{1}\) and the ICM's \(O_{2^{\prime}}\) would be the accuracy improvement achieved. The higher the resolution factor \(S\), the greater the improvement in accuracy. Figure 9: Compass Method error for a 1 m subdivision mesh. The ICM is more accurate the higher the \(S\). It is perfectly suitable for a single meshing process but has an important drawback when performing an optimisation problem, as it requires much more memory to compute the temporary intersection points over a surface, which slows down the algorithm drastically. The additional temporary intersection points necessary to obtain a quad mesh can be calculated according to:  
\[\text{Additional intersections}=(S+1)^{2}-\text{ number of steps} \tag{9}\]  
The performance of both methods is analysed on the test surface. The CM and the ICM are configured with the same radius, and the latter with \(S=3\). According to the results shown in Figure 11, the ICM needs to compute 15 new points to obtain the rest of the three vertices of a new quad mesh (\(O_{0}\)-3), 12 more than the CM. These points are temporal or instrumental intersections. For a single mesh, this difference would have no practical impact, but for an optimisation problem with a few hundred runs, if not thousands, it is a key performance issue to consider. To overcome this drawback, the authors have developed the Adaptative Compass Method (ACM). Figure 6:0 Compass Method (**a**) and Improved Compass Method (**b**) comparison. Figure 6:1 Improved Compass Method versus Compass Method.