A relative-geometric treatment of ruled surfaces

Abstract

We consider relative normalizations of ruled surfaces with non-vanishing Gaussian curvature $K$ in the Euclidean space $\mathbb{R} ^{3}$, which are characterized by the support functions $^{\left( \alpha \right) }q=\left \vert K\right \vert ^{\alpha}$ for $\alpha \in \mathbb{R}$ (Manhart's relative normalizations). All ruled surfaces for which the relative normals, the Pick invariant or the Tchebychev vector field have some specific properties are determined. We conclude the paper by the study of the affine normal image of a non-conoidal ruled surface.Comment: 12 page

On ruled surfaces relatively normalized

“…By combining this last relation with (10) and (38) we get x = c ȳ + ā, which means that Φ is a proper relative sphere. Thus, we arrive at Proposition 5 An asymptotically normalized ruled surface Φ is a proper relative sphere iff the function f is given by (36) and its fundamental invariants are related as in the equation (37).…”

“…In this paper only skew ruled surfaces of the space E 3 are considered with parametrization like in (10) and (11).…”

“…is its striction curve and H 1 is the relative mean curvature of Ψ 1 . Thus Ψ 2 is parametrized like in (10) and (11). Obviously, the Tchebychev vector T1 of ȳ1 is parallel to ē.…”

“…is its striction curve and H i−1 is the relative mean curvature of Ψ i−1 . Ψ i is parametrized like in (10) and (11) and its fundamental invariants are computed by…”

On the Laplace Normal Vector Field of Skew Ruled Surfaces

“…Relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the Euclidean space R 3 have been studied in the last years in many points of view (see [2], [4], [5], [6]; for more details and references see [8]). This paper deals with the Laplace normal vector field of skew ruled surfaces.…”