We consider a skew ruled surface $\Phi$ in the Euclidean space $E^{ 3}$ and relative normalizations of it, so that the relative normals at each point lie in the corresponding asymptotic plane of $\Phi$. We call such relative normalizations and the resulting relative images of $\Phi$ \emph{ asymptotic}. We determine all ruled surfaces and the asymptotic normalizations of them, for which $\Phi$ is a relative sphere (proper or inproper) or the asymptotic image degenerates into a curve. Moreover we study the sequence of the ruled surfaces ${ \Psi_{ i}}_{ i\in \mathbb{ N}}$, where $\Psi_{ 1}$ is an asymptotic image of $\Phi$ and $\Psi_{ i}$, for $i\geq2$, is an asymptotic image of $\Psi_{ i-1}$. We conclude the paper by the study of various properties concerning some vector fields, which are related with $\Phi$.
Categories: math.DG