Ioanna-Iris Papadopoulou, Stylianos Stamatakis

On polar relative normalizations of ruled surfaces

This paper deals with skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$ which are equipped with polar normalizations, that is, relative normalizations such that the relative normal at each point of the ruled surface lies on the corresponding polar plane. We determine the invariants of a such normalized ruled surface and we study some properties of the Tchebychev vector field and the support vector field of a polar normalization. Furthermore, we study a special polar normalization, the relative image of which degenerates into a curve.

Keywords:

Pick invariant, Polar normalizations Ruled surfaces, Tchebychev vector field,

PDF

___

[1] F. Manhart, Eigentliche Relativsph¨aren, die Regelfla¨chen oder Ru¨ckungsfla¨chen sind, Anz. O¨ sterreich. Akad. Wiss. Math.-Natur. Kl. 125 (1988), 37–40.
[2] F. Manhart, Relativgeometrische Kennzeichnungen Euklidischer Hypersph¨aren, Geom. Dedicata 29 (1989), 193–207.
[3] H. Pottmann and J. Wallner, Computational Line Geometry, Springer-Verlag, New York, 2001.
[4] P. A. Schirokow and A. P. Schirokow, Affine Differentialgeometrie, B. G. Teubner Verlagsgesellschaft, Leipzig, 1962.
[5] S. Stamatakis and I. Kaffas, Ruled surfaces asymptotically normalized, J. Geom. Graph. 17 (2013), 177–191.
[6] S. Stamatakis, I. Kaffas and I.-I. Papadopoulou, Characterizations of ruled surfaces in R3 and of hyperquadrics in Rn+1 via relative geometric invariants, J. Geom. Graph. 18 (2014), 217–223.
[7] S. Stamatakis and I.-I. Papadopoulou, On ruled surfaces relatively normalized, Beitr. Algebra Geom. 58 (2017), 591–605.
[8] S. Stamatakis and I.-I. Papadopoulou, Ruled surfaces right normalized, ArXiv:1706.07277 [math.DG].
[9] G. Stamou and A.Magkos, Regelefl¨achen relativgeometrisch behandelt, Beitr. Algebra Geom. 45 (2004), 209–215.
[10] G. Stamou, S. Stamatakis and I.Delivos, A relative-geometric treatment of ruled surfaces, Beitr. Algebra Geom. 53 (2012), 297–309.