On polar relative normalizations of ruled surfaces

Abstract

This paper deals with skew ruled surfaces in the Euclidean space 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.

On the trajectory ruled surfaces of framed base curves in the Euclidean space

“…So trajectory ruled surfaces are one of the most important topics of ruled surface theory to study spatial mechanism and space kinematic. For more details on ruled surfaces and its applications, see previous studies 1–6 …”

“…For more details on ruled surfaces and its applications, see previous studies. [1][2][3][4][5][6] In differential geometry, the curve theory is highly important and a classical subject. Generally, geometers consider regular curve, if they work on curve theory.…”