There exists a natural correspondence between null curves in $\mathbf{C}^4$ and "free" curves on $\mathcal O(1)\oplus \mathcal O(1)$; it underlies the existence of "Weierstrass type formulae" for minimal surfaces in $\mathbf{R}^4$. The construction determines correspondences for minimal surfaces in $\mathbf{R}^3$, and constant mean curvature 1 surfaces in $\mathrm{H}^3$; moreover it facilitates the study of symmetric minimal surfaces in $\mathbf{R}^4$.
