Journal for Geometry and Graphics 17 (2013), No. 2, 177--191
Copyright Heldermann Verlag 2013
Ruled Surfaces Asymptotically Normalized
Stylionos Stamatakis
Dept. of Mathematics, Aristotle University, 54124 Thessaloniki, Greece
stamata@math.auth.gr
Ioannis Kaffas
Dept. of Mathematics, Aristotle University, 54124 Thessaloniki, Greece
[Abstract-pdf]
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 $\Phi$ 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 $\{\Psi_{i}\}_{i\in N}$, where $\Psi_1$ is an asymptotic image of $\Phi$ and $\Psi_i$, for $i\geq 2$, 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 $\Phi$.
Keywords: Ruled surfaces, relative normalizations.
MSC: 53A25; 53A05, 53A15, 53A40
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