On the Laplace Normal Vector Field of Skew Ruled Surfaces

Abstract

We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space $\mathbb{R}^{3}$. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve. Moreover, we study the Laplace normal image of a non-conoidal ruled surface whose relative normals lie on the asymptotic plane.