# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 2 Preliminary Theoretical Considerations
It must be noted that the position of the axial and cusp tangent lines and, consequently, the corresponding resistance areas depend on \(\alpha\). Obviously, a cylindrical abutment (zero taper) will provide the maximum axial resistant area but no cusp resistance. Therefore, \(\alpha\) is necessary to be appropriately sized [17; 28]. Occlusal forces can have different application points and different directions on occlusal surfaces. The direction is called _horizontal_ when the points of the applied force are located at the level of the occlusal walls of the crown, therefore above and outside the occlusal surface of the abutment. The direction orthogonal to the first one is called _vertical_, and, along this direction, two cases can be distinguished:  
Figure 1: Geometry of the abutment with cusp. * Force within the perimeter of the abutment: The prosthetic crown is not subjected to any dislodging rotation. The effects of the force are due to the component along the axial wall of the abutment (called the _sliding force_\(F_{s}\)) and to the component orthogonal to it (called the _compression force_\(F_{c}\)) (Figure 2). When \(\alpha=45^{\circ}\), the two components are equal. * Force external to the perimeter of the abutment: A lever is generated with a fulcrum in the cervical margin point (\(F_{1}\)) of the preparation. Thus, the dislodging moment generated is contrasted by the resistance areas opposed by means of two moments having arms \(r_{\rm a}\) and \(r_{\rm c}\), respectively. As shown in Figure 3, \(b_{\rm F}\) is the dislodging arm, while \(r_{\rm a}\) and \(r_{\rm c}\) are the resistance arms. The geometric analysis of the resistance area of the axial surface [28] is based on two preparation morphological parameters (Figure 4): the width (\(b\)) and the length of the segment joining the fulcrum (\(F_{1}\)) with the contralateral cusp vertex (\(V_{C1}\)), forming an angle \(\beta\) with the axial wall. The \(\delta\) angle is the axial wall slope. Figure 3: Occlusal force external to the base perimeter of the abutment. For obvious geometric reasons, the fulcrum-contralateral cusp vertex segment (\(F_{1}V_{c1}\)), in the case of a _cylindrical__subtent_ (\(\alpha=0^{\circ}\)), is always greater than the base (\(b\)), whatever the axial surface height. Therefore, under the action of a force parallel to the occlusal plane, the entire axial surface constitutes a resistance area. In the case of a conical abtment (\(\alpha\
eq 0^{\circ}\)), the three possibilities that can occur are as follows:  
* \(F_{1}V_{c1}=b\): the axial resistance area is equal to half of the contralateral axial surface;
* \(F_{1}V_{c1}<b\): the axial resistance area is less than half of the contralateral axial surface;
* \(F_{1}V_{c1}>b\): the axial resistance area is greater than half of the contralateral axial surface. The geometric analysis of the resistance area of the cusp surface [28] is also based on the morphological parameters' preparation. With reference to a _conical abtment_ (Figure 5), the three cases that can occur are as follows:  
* \(F_{1}V_{c2}=F_{1}C\): the cusp resistance area, as the height of the cusp groove varies, is equal to half of the cusp surface;
* \(F_{1}V_{c2}>F_{1}C\): as the height of the cusp groove decreases, keeping the base width constant, the cusp resistance area increases;
* \(F_{1}V_{c2}<F_{1}C\): the cusp resistance area is less than half the area of the cusp wall. Figure 4: Morphological parameters of the preparation for the analysis of the axial resistance area. Figure 5: Morphological parameters of the preparation for the analysis of the cusp resistance area.