## 3 Results and Discussion
This result indicates that the Anterior SC is the most curved canal, and also explains the behavior of the AP and AH angles, which strongly depend on the applied threshold. Even though the separations we obtained for a rejection threshold of 40% are \(\langle s_{H}\rangle=0.08\) mm, \(\langle s_{A}\rangle=0.15\) mm, and \(\langle s_{P}\rangle=0.10\) mm, where \(\langle\cdot\rangle\) denotes the average computed over the whole group of volunteers, individual separations can be as large as \(s_{H}=0.36\) mm, \(s_{A}=0.34\) mm and \(s_{P}=0.36\) mm for the whole canal and \(s_{H}=0.17\) mm, \(s_{A}=0.24\) mm and \(s_{P}=0.17\) mm for 40% rejection, respectively. Figure 10: Standard deviation for the A, P and H plane separation from the reference plane as a function of the percentage of discarded SC. Grey lines represent different volunteers, the black line is the average and the red lines represent the limits for one standard deviation for each SCs. Figure 9: A, P and H non-planarity (defined as normal distance) referred to the reference plane: (**left**) this distance when the whole duct is retained; and (**right**) the evolution of this distance when the 40 th percentile is discarded. Grey lines represent different volunteers, the black line is the average and the red lines represent the limits for one standard deviation for each SCs. [PERSON] et al. [12] described the bending of the SCs ducts assuming that they lie on a curved surface (a generalized cylinder) instead of a plane. From the data published in the Appendix of that paper, we could estimate that the non-planarity for their model assigns a standard deviation of the distance to the respective planes of \(s_{A}=0.29\) mm, \(s_{P}=0.19\) mm and \(s_{H}=0.16\) mm, values which are compatible with the values we obtained using a rejection of roughly 5% (see the thick black lines on Figure 10). From our data for posterior and horizontal SCs, we observe that a slight variation on the decision about where are the ampulae located could modify the parameters of this model. Again, the decision on how the ampulae is considered can affect the parameter values of any model. As in previous works, we verified that the centerlines are very close to ellipsoidal trajectories, thus we extracted the parameters that can be obtained from the individual paths. A least square fitting was done (based on an algebraic distance) for each volunteer and canal. Figure 11 shows in red the ellipse whose semi-axes are the average of the whole set of ellipses that better describe each of the SCs projected on its corresponding plane. We present two situations, retaining all the duct or with a rejection threshold of 40%. The individual trajectories appear in gray in the same figure. It must be noted that the ampulae position, in most of the cases, is located at one end of the major axis. We can appreciate small differences of the fitting parameters of the ellipses in both 0% and 40% rejection thresholds. In Table 4, we present the variability of the semi-axes. In both cases, our data include the average and their corresponding standard deviation for the whole set of volunteers. We include for reference the values published in two previous works, by [PERSON] et al. [12] and [PERSON] et al. [24]. The values from this last study were inferred from their published values, as they only included the size of the circular cavity defined inside each canal. As our characterization is done using the central path along the ducts, we enlarged their values with the canal radii values of our work. We can appreciate that there is no real difference between these three studies, as the variability (defined from the standard deviation in our cohort) of these parameters is a 10%. However, it should  
Figure 11: Centerlines projected on the respective reference planes for the A, P and H ducts: (**top**) the behavior for the whole ducts; and (**bottom**) when 40% is discarded. The red thick line correspond to the ellipse that best fits all the data from the different volunteers.