input: relative angles $〈n_i,n_j〉$ for three mirror positions $\mathcal{M}\left(n_1\right),\mathcal{M}\left(n_2\right),\mathcal{M}\left(n_3\right)$, and the coordinates of the corresponding laser reflections: $\pi \left(R\left(n_1\right)\right),\pi \left(R\left(n_2\right)\right),\pi \left(R\left(n_3\right)\right)$. query: $n_4$, or rather $〈n_i,n_4〉$ for some $i=1,2,3$. output: $\pi \left(R\left(n_4\right)\right)$.