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.

arXiv commentappears in the Proceedings of the 1st International Workshop on Line Geometry and Kinematics (IW-LGK-11), April 26-30, 2011, Paphos, Cyprus