## 4 Results of Modeling and Simulations: Shapes of the Scan Patterns  
Using a commercially-available mechanical design program, CATIA5V20, by considering finite rotations of the two prisms shown in Figure 1a, the ray bundles can be obtained, as demonstrated in Figure 1b,c. Therefore, by intersecting them with a certain plane (i.e., the screen positioned at the distance \(L\) from the final diopter perpendicular on the O.A.--Figure 2), the generated scan patterns can be obtained. In the following part of the study, mostly the case of the scanner configuration in Figure 2d is considered to carry out simulations, while a comparison of the scan patterns of all four configurations in Figure 2a-d is also provided. To better understand the kinematics of the laser spot, in Figure 3 the case when one of the prisms is fixed and the other one is mobile is considered. Thus, for four different positions rotated with \(90^{\circ}\) of the fixed prism, the rotation of the other prism producesfour circles in the scanned plane--one for each position of the fixed prism. Because of the different longitudinal position of the two prisms (i.e., along the O.A. ), the circles obtained when Prism 1 is mobile and Prism 2 is fixed are larger, as Prism 1 is situated at a longer distance from the screen. By combining the two rotational movements of the prisms, a Rhononea-type curve is obtained, albeit with different equations than such classical curves [62], as shown initially in Figure 1 for the same \(|M|\), but with an opposite (Figure 1c) versus an identical (Figure 1b) rotational sense. Examples of such scan patterns, with a multi-parameter analysis carried out considering the \(M\) and \(k\) parameters in Equation (1), as well as the other constructive parameters of the scanners are provided in Figures 4-8. In Figure 4, the scan patterns are generated for the most common case of identical prisms, i.e., of \(k=1\) (obtained for \(D_{1}=D_{2}=2^{\circ}\)) and several values of \(M\), positive on column 1 and negative on column 2: (a) \(|M|=2\); (b) \(|M|=4\); (c) \(|M|=6\); (d) \(|M|=8\). The \(|M|=4\) case corresponds to the constructions in Figure 1.  
One can remark that the number of loops of a pattern is \(\mu=M-1\) for \(M>0\) (e.g., \(\mu=1\) for \(M=2\), \(\mu=2\) for \(M=3\), etc.). The difference between positive and negative values of \(M\) can be seen, as \(\mu=|M|+1\) for \(M<0\) (e.g., \(\mu=3\) for \(M=2\), \(\mu=4\) for \(M=3\), etc.). From this comparison, the number of loops is in general, as first observed in [4],  
\[\mu=|M-1|\,, \tag{37}\]  
Figure 3: Circles obtained in the scanned plane when one of the prisms is fixed (positioned in four successive positions, at \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), and \(270^{\circ}\)) and the other one is mobile. The four larger circles correspond to the situation when Prism 1 (positioned further away from the screen than Prism 2) is mobile. The values on the axes are in millimeters. The first number represents the rotational angle of Prism 1 and the second that of Prism 2, for example 0:0... 360 means that Prism 1 is fixed (in the \(0^{\circ}\) position) and that Prism 2 rotates from 0 to \(360^{\circ}\); 0... 360:90 means that Prism 1 performs a full rotation while Prism 2 is fixed at \(90^{\circ}\) from its \(\varphi=0\)) position in Figure 2(d1). Figure 4: Study of the scan patterns produced by a pair of rotational Risley prisms (i.e., the configuration \(\mathit{bu}\)-\(\mathit{ab}\) in Figure 2d) for \(L=1\) m, \(\varepsilon=25\) mm and for: (**a**) \(|M|=2\); (**b**) \(|M|=4\); (**c**) \(|M|=6\); (**d**) \(|M|=8\). (**1**) Left column, \(M>0\); (**2**) right column, \(M<0\). The \(k=1\) case was considered for identical prisms with an individual deviation angle \(D=(n-1)\theta=2^{\circ}\). The values on the axes are in millimeters. Figure 5: Study of the scan patterns produced by the configuration _ba-ab_ of rotational Risley prisms (Figure 2d), for \(L=1\) m, \(e=25\) mm, and \(|M|=6\), with: (**a**) \(k=1/2\); (**b**) \(k=2\); (**c**) \(k=3\); (**d**) \(k=5\).