## 4 Results and Discussion
### Geometric Comparison of the Obtained Models  
With the objective of analyzing the geometric behavior of the obtained models, a comparison was made between the models that are generated with three methods: (a) using a fixed Bezier degree for all of the graphite nodules; (b) the proposed methodology; and, (c) with a commercial software, img2 CAD. There are different works that used this commercial software to generate their models in a CAD format [34; 35]. Table 1 shows the models obtained from four nodules (A, B, C and D) using the three different methods mentioned, besides the \(CSF_{\text{est}}\) and \(E_{\text{r}}\) values for each graphite nodule are shown, where the maximum errors are highlighted in bold. Regarding the models that are generated using a fixed Bezier degree for all the nodules, three different models were created with the following degrees: 4, 10, and 18. Figure 7: Contours of the 14 graphite nodules: at pixel level obtained by image processing (black) and model curves obtained by genetic algorithms (red). Table 1 shows the generated models, where geometric evolution it can be seen as the Bezier curve degree increases. In relation to the models obtained using a fixed Bezier curve degree for all of the nodules, it can be seen that the curves that are obtained with Bezier degrees of 4 are curves with high \(CSF_{\text{est}}\) values and high \(E_{\text{r}}\) values, therefore do not approximate adequately to the geometry of the model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Model** & **Nodule A** & **Nodule B** & **Nodule C** & **Nodule D** \\ \hline \(n=4\) & & & & \\ \(CSF_{\text{est}}\) & 0.9911 & 0.7820 & 0.9930 & 0.9260 \\ \(E_{\text{r}}\) & 0.1203 & **0.7019** & 0.5465 & **0.8376** \\ \hline \(n=10\) & & & & \\ \(CSF_{\text{est}}\) & 0.8902 & 0.6239 & 0.9348 & 0.7879 \\ \(E_{\text{r}}\) & 0.0062 & 0.3578 & 0.4559 & 0.5635 \\ \hline \(n=18\) & & & & \\ \(CSF_{\text{est}}\) & 0.1231 & 0.5158 & 0.8858 & 0.7871 \\ \(E_{\text{r}}\) & **0.8608** & 0.1225 & 0.3796 & 0.5657 \\ \hline \((b)\) & & & & \\ \(n_{\text{opt}}\) & 9 & 13 & 20 & 24 \\ \(CSF_{\text{est}}\) & 0.9247 & 0.5764 & 0.884 & 0.7275 \\ \(E_{\text{r}}\) & 0.0452 & 0.2544 & 0.3769 & 0.4437 \\ \hline \((c)\) & & & & \\ \(CSF_{\text{est}}\) & 0.8532 & 0.46722 & 0.6291 & 0.5020 \\ \(E_{\text{r}}\) & 0.0354 & 0.0168 & 0.0201 & 0.0037 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric comparison of the obtained models using the three different methods: (a) using a fixed Bézier degree for all the graphite nodules; (b) the proposed methodology; and (c) with a commercial software, img2 CAD. digital contours; on the other hand, as the curve degree increases (degree 10), the curves approximate to the contours obtained digitally preserving the smoothness that characterizes them, unlike the models that are obtained with commercial img2 CAD software and the models that are obtained by another methodologies [11; 14; 16], where no smooth algorithms has been applied. It can also be observed for the models with a curve degree of 10, that the \(CSF_{\text{est}}\) values decrease noticeably as opposed to the \(CSF_{\text{est}}\) values that are obtained with degree 4, as a better approximation with respect to the digital contours is obtained. When it is used a degree of 18, depending of the analyzed nodule, geometries with some acute contours are generated (nodule B), which causes the \(CSF_{\text{est}}\) value to decrease even more than the obtained ones with a degree of 10 due to the great similarities with the digital contours and the repeatability of geometric errors that are created by the discretization process. Another feature that can be seen is that nodules B, C, and D can be represented with curves degrees of 18 without any problem, unlike smaller nodules like A. This effect can be explained as the system represented in Equation (2) becomes indeterminate as the curve degree used is greater than the number of points that the contour has, generating spurious geometries (blue contour).