Ruled surfaces asymptotically normalized

Abstract

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 $i\ge 2$, 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 $\Phi$.

arXiv comment15 pages

Cited by in corpus (2)

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