## 2 Proposed Inverse Solution for Risley Prisms
### Geometric Method, Circumference Approximation  
In this first approximation, Figure 4a, the geometric shape that the beam describes when the second prism \(\pi_{2}\) rotates approximates a circumference \(c_{2}\) with centre \(C\) at any point of the circle \(c_{1}\). The circumference \(c_{1}\) (with the centre at the point of origin, \(O\)) is described by the beam when the first prism \(\pi_{1}\) rotates and there is no second prism \(\pi_{2}\). Using the Forward Vector Refraction Theorem, the maximum and minimum radius that the system can reach at a given distance \(D2\) can be calculated. The maximum radius (\(r_{max}\)) is obtained when both prisms are oriented with the same angle of rotation and the minimum radius (\(r_{min}\)) is obtained when the prisms are \(180^{\circ}\) out of phase. Once both values have been calculated, the primary and secondary radius (\(r_{1}\) and \(r_{2}\)) can be obtained.  
\[\begin{split} r_{1}&=r_{2}+r_{min}\\ r_{2}&=(r_{max}-r_{min})/2\end{split} \tag{3}\]  
Figure 3: Risley system diagram.  
Given the target point P to be reached with coordinates \((x_{0},\ y_{0})\), a change of axes through the angle \(\delta\) is made, Figure 4b, so that the target point P is on the X\({}^{\prime}\) axis and now becomes P\({}^{\prime}\) (\(x_{0}^{\prime},y_{0}^{\prime}\)).  
Considering \(y_{0}^{\prime}=0\) and solving the system of equations in (4) that define the radii \((r_{1},\ r_{2})\), the centre point \(C^{\prime}(x_{c}^{\prime},y_{c}^{\prime})\) is obtained.  
\[\begin{split} r_{1}=\sqrt{x_{c}^{\prime 2}+y_{c}^{\prime 2}}\\ r_{2}=\sqrt{(x_{c}^{\prime}-x_{0}^{\prime})2+y_{c}^{\prime 2 }}\end{split} \tag{4}\]  
Knowing the centre point \(C^{\prime}\), it is possible to calculate the angle \(\gamma^{\prime}\):  
\[\gamma^{\prime}=acos\big{(}x_{c}^{\prime}/r_{1}\big{)}=acos\Big{(}\Big{(}r_{1 }^{2}-r_{2}^{2}+{x_{0}^{\prime}}^{2}\Big{)}/2r_{1}x_{0}^{\prime}\Big{)} \tag{5}\]  
Finally, undoing the change of coordinate axes, the rotation angle of the first prism \(\theta_{1}\) is:  
\[\theta_{1}=\gamma^{\prime}+\delta \tag{6}\]  
where \(\delta\) is:  
\[\delta=acos\bigg{(}x_{0}/\sqrt{x_{0}^{2}+y_{0}^{2}}\ \bigg{)} \tag{7}\]  
The centre point of the circumference \(C^{\prime}(x_{c}^{\prime},\ y_{c}^{\prime})\) on the X\({}^{\prime}\) and Y\({}^{\prime}\) coordinate axes is isolated from (5) to obtain the rotation angle of the second prism \(\theta_{2}\):  
\[\begin{split} x_{c}^{\prime}&=r_{1}\cdot cos(\gamma ^{\prime})\\ y_{c}^{\prime}&=r_{1}\cdot sin(\gamma^{\prime}) \end{split} \tag{8}\]  
Translating the centre point C\({}^{\prime}\) to the X and Y coordinate axes:  
\[\begin{split} x_{c}&=x_{c}^{\prime}\cdot\cos( \delta)-y_{c}^{\prime}\cdot\sin(\delta)\\ y_{c}&=x_{c}^{\prime}\cdot\sin(\delta)+y_{c}^{ \prime}\cdot cos(\delta)\end{split} \tag{9}\]  
The rotation angle of the second prism \(\theta_{2}\) is the angle of the section formed by the points \((C,\ P,\ O_{2})\) in Figure 4a.  
\[\theta_{2}=acos\bigg{(}(x_{0}-x_{c})/\sqrt{\left(-x_{0}+x_{c}\right)^{2}+ \left(-y_{0}+y_{c}\right)^{2}}\ \bigg{)} \tag{10}\]  
Figure 4: Rotation of the coordinate axes of circumference approximation. (**a**) represents the original axes (X, Y); (**b**) represents the rotated axes (X\({}^{\prime}\), Y\({}^{\prime}\)) though the angle \(\delta\).