Range-Azimuth-Elevation

Common model for sensors like LiDAR

$$\mathbf{\rho }={\mathbf{r}}_s^{ps}=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$$

The sensor itself gets readying by RAE, but we get it in 3D points

$$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}r\cos \alpha \cos ϵ\\ r\sin \alpha \cos ϵ\\ r\sin ϵ\end{array}\right]$$

This is the inverse of the sensor model, so the sensor model is actually

$$\left[\begin{array}{c}r\\ \alpha \\ ϵ\end{array}\right]=\mathbf{s}(\mathbf{\rho })=\left[\begin{array}{c}\sqrt{x^2+y^2+z^2}\\ {\tan }^{-1}\left(\frac{y}{x}\right)\\ {\sin }^{-1}\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)\end{array}\right]$$

For 2D lidars, we set $z=0$