## 2 Materials and Methods  
### Geometrical Core Concepts  
In order to highlight the tooth volume, the three-dimensional Cartesian coordinate system, with the origin in (O) and with the axes (Op), (Oq), and (Or), oriented as shown by the drawn arrows in Figure 1, is defined. The fictional axis with the (Or\({}_{\text{f}}\)) length (figured with red line) is associated with the elliptic parabolic (with (P) and (Q) radii) expressed by:  
\[\frac{p^{2}}{P^{2}}+\frac{q^{2}}{Q^{2}}=2\cdot r_{f} \tag{1}\]  
Which, in the assumption that \(\text{p}=\text{P}\) and \(\text{q}=\text{Q}\), corresponds to the height \(\text{r}_{\text{f}}=1\). In Figure 1a, the upper central incisor is presented in its normal position in the tissue, highlighting the fact that its form can be approximated with high accuracy with an elliptic parabolic. In Figure 1b, the same upper central incisor is presented rotated at 1800. This representation is used in order to \"normalize\" the parabolic position in order for its peak to correspond with the origin of the Cartesian system. This procedure does not change the physical evolution and applicability of the orthodontic treatment, but it simplifies the treatment presentation manner, which becomes a more intuitive one. In Figure 1c, only the approximating elliptic parabolic, for the considered upper central incisor, is presented. Next, the tooth properties are computed using the geometrical properties of the approximating elliptic parabolic. It can be easily observed that the parabolic volume (V) is equal to:  
\[V=\frac{1}{2}\cdot\pi\cdot P\cdot Q\cdot r_{f}, \tag{2}\]  
In the scenario that a tooth has P = 2 mm, Q = 3 mm, and height r\({}_{\text{f}}\) = 24 mm, its volume will be equal to: V = \(\frac{1}{2}\cdot\pi\cdot 2\cdot 3\cdot 24\cong 226.2\) mm\({}^{3}\). For the fictional shaft presented in Figure 1, having the Or\({}_{\text{f}}\) = r\({}_{\text{f}}\) length, in Figure 2 we associate the following notations:  
\[r_{f}=s_{f}^{\prime}=s_{f}+\tilde{s}^{\prime}+\tilde{s} \tag{3}\]  
Figure 1: Schematic representation of the upper central incisor, which is the subject of the orthodontic movement and of the elliptic approximating parabolic [21]. (**a**) The upper central incisor in normal position. (**b**) The upper central incisor rotated by 180\({}^{\circ}\). (**c**) The elliptic parabolic which is used for the approximation of the upper central incisor surface. Moreover, \(y_{\beta 0}=0\) represents the tooth apex (tip of the tooth root) and (\(y^{\prime}_{a0}\)) is the visible part of the tooth crown-incisal edge, both in the initial position. The overall tooth length, represented with the following statement:  
\[s_{f}=s^{\prime}_{f}-(\widetilde{s}+\widetilde{s}^{\prime}), \tag{4}\]  
is considered to be located inside the plastic environment (inside the tissue) subject to deformation. Additionally, the actuation point where the driving force \(\mathfrak{u}_{0}=\mathfrak{u}_{0}(t)\), generated by the elastomeric chain, \(\mathfrak{u}_{0}=\mathfrak{u}_{0}(t)\), is applied, is considered to be at the distance (\(\mathfrak{s}_{\mathrm{f}}+\ \widetilde{\mathfrak{s}}\)) from the tip of the tooth root (apex). As a consequence of the orthodontic treatment (of the \(\mathfrak{u}_{0}=\mathfrak{u}_{0}(t)\) force application) during a period of time \(t=\mathfrak{t}_{f}\), on the visible part of the tooth's incisal edge will appear a position modification having the value (\(+\mathfrak{y}^{\prime}_{\alpha}\)). The fictional axis of the tooth can describe either rototranslation or translational movements, cases in which the tooth apex shows a shift from the current position \(\mathfrak{y}_{\beta 0}=0\) to negative (\(-\mathfrak{y}_{\beta}\)) or positive (\(+\mathfrak{y}_{\beta}\)) values.