## 5 Multi-Parameter Analysis of Angular and Linear Deviations  
### Multi-Parameter Analysis of Angular Deviations  
The first step of the analysis refers to the angular deviations of the four scanners configurations presented in Figure 2, with expressions given by Equations (2) to (5). Figure 9a shows the maximum angular deviations \(D_{j}^{max}\), \(j=a\), \(b\), \(c\), \(d\) as functions of the angle of the prisms--for the case of identical prisms, therefore with \(n_{1}=n_{2}=n\) and \(\theta_{1}=\theta_{2}=\theta\) (i.e., with \(k=1\)). These exact values of deviations are compared with the approximate/linearized one, \(D_{lin}^{max}\), given by Equation (7). One can remark that the latter is the same regardless of the scanner configuration. Figure 9b provides the error between the case when the maximum angular deviations are considered in their linear approximation, instead of their exact values. One can see that this approximation can be roughly valid up to a prism angle of \(0.087\) rad = \(10^{\circ}\), and therefore for optical wedges. For prisms with a larger angle, one must consider the exact expressions of the deviations--angular as well as linear. Another conclusion that can be extracted from the study in Figure 9b is that, from Figure 2, the configurations (a) and (d) have much higher non-linearities than configurations (b) and (c). The strongest non-linearity refers to configuration (a), while the weakest one characterizes configuration (c). From Figure 9 one can see that a limit prism angle, \(\theta_{j}^{lim}\), \(j=a\), \(b\), \(c\), \(d\), can be determined for each scanner configuration, as a function of the refractive index \(n\) of the prisms, as remarked from Figure 10 as well. Figure 10: Maximum angular deviations, \(D_{j}^{max}\), \(j=a\), \(b\), \(c\), \(d\), as functions of the angle \(\theta\) of identical prisms for two values of the refractive index: \(n=1.517\) and \(n^{\prime}=1.7\). Each scanner configuration, (**a–d**) is considered separately, while the exact angular deviations are compared with the approximate/linearized one for each refractive index. Figure 9: (a) Maximum angular deviations \(D_{j}^{max}\), \(j=a\), \(b\), \(c\), \(d\) as functions of the angle \(\theta\) of identical prisms, presented in comparison to the approximate/linearized one, \(D_{lim}^{max}\), Equation (7); (b) the absolute error between the maximum angular deviations and their linear approximation. A refractive index \(n=1.517\) was considered for both prisms. From Figure 2a this limit prism angle is obtained from the conditions to avoid total reflections on the second and fourth diopter of the scanner. These conditions are  
\[\epsilon_{2}^{\prime}<\pi/2\text{ and }\epsilon_{4}^{\prime}<\pi/2,\text{ respectively}. \tag{39}\]  
By using each scanner equation in Table 1, these inequations can give limit values of the angles of each prism, \(\theta_{1}\) and \(\theta_{2}\). For the particular case of identical prisms and using the small angles approximation, these inequations can then provide an approximate limit value of \(\theta_{j}^{lim}\), \(j=a\), \(b\), \(c\), \(d\). It is important to remark from Figures 9 and 10 that such limits are higher than the exact ones, which are smaller when the Gaussian approximation is not utilized.