Oriented straight lines and twistor correspondence

The tangent bundle to the $n$--dimensional sphere is the space of oriented lines in $\R^{n+1}$. We characterise the smooth sections of $TS^n\to S^n$ which correspond to points in $\R^{n+1}$ as gradients of eigenfunctions of the Laplacian on $S^n$ with eigenvalue $n$. The special case of $n=6$ and its connection with almost complex geometry is discussed.