## 1 Introduction  
Beam scanners make it possible to obtain the geometry of a scanned environment from massive data sets. Several beam scanners are available for this purpose, including gimbal, galvanometer, and Risley prism types. While traditional carried-axis gimbal scanners can be heavy and consequently require more driving energy and non-carried-axis galvanometers have limited aperture size, Risley scanners can offer a lighter, more compact, vibration-insensitive scanning option. The study described in this paper is based on Risley prism scanners. The Risley configuration consists of a combination of two or more wedge prisms [1] that can rotate independently, Figure 1. The system allows the direction of the beam \(\stackrel{{\rightarrow}}{{S}}\) to be set to a specific point P in space, and different patterns can be generated by setting the relative velocity between the rotating prisms. Among other uses, the Risley configuration has been applied in LiDAR technology [2, 3] to scan the surface of the Earth with different scanning patterns or to collect wind field  
Figure 1: Risley system diagram in 3D. data [4] from an aircraft. The Risley configuration allows the development of systems with a wide field of view, multi-target tracking, and angular scanning with digital micromirror devices (DMDs) [5]. Two problems have arisen in beam steering applications of Risley prism systems, the forward and the inverse problem [6]. The forward problem, i.e., determine the deviation of the laser beam direction \(\left(\overset{\rightarrow}{A_{rf}}\right)\) given a set of prism orientation angles, has been solved by the Forward Vector Refraction Theorem [7]:  
\[\overset{\rightarrow}{A_{rf}}=n\overset{\rightarrow}{A_{r3}}+\left\{\sqrt{1- n^{2}\cdot\left[1-\left(\overset{\rightarrow}{A_{r3}}\cdot\overset{ \rightarrow}{N_{22}}\right)^{2}\right]}-n\overset{\rightarrow}{A_{r3}}\cdot \overset{\rightarrow}{N_{22}}\right\}\cdot\overset{\rightarrow}{N_{22}}=\left( x_{rf},\,y_{rf},\,z_{rf}\right)^{T} \tag{1}\]  
The inverse problem is based on calculating the rotation angles of the two prisms from the final emerging vector \(\left(\overset{\rightarrow}{A_{rf}}\right)\). The final emerging vector at a known distance \(D2\) is a specific coordinate. In other words, given a target coordinate at a distance of \(D2\), the inverse problem is to calculate the rotation angles of the two prisms. Several approaches to the inverse problem have been published by different researchers: an iterative process, which presents a high precision solution [7], the third-order solution [8], the two-step method [9; 10; 11], the look-up table method [12], the control system based on [PERSON]'s method [13], and the error compensation method based on the paraxial approximation [14], among others. However, these approaches have a number of drawbacks, including a lack of accuracy in many critical cases and/or high computation time. The work presented in this study greatly increases the accuracy of existing approaches. To achieve a less complicated solution that can be used in applications requiring fast response and high accuracy, a geometry-based method has been developed which aims to determine the equations that define the geometric shapes described by the final beam when it passes through the second prism \(\pi_{2}\) in a plane at a specific distance \(D2\). On the basis of these equations, the intersection points with the circumference described by the beam when it passes through the first prism \(\pi_{1}\) of the system can first be calculated, and then, knowing these intersection points, the angles of rotation of the desired prisms can be calculated. This process is the focus of our research and has been called the 'geometric method'. Varying the aperture angle of the prisms and using the Forward Vector Refraction Theorem, it can be seen in Figure 2 that the beam describes a deformed ellipse when the second prism of the Risley system makes a full rotation. This deformation is more severe as the wedge angle of the prism is increased.