On the Laplace Normal Vector Field of Skew Ruled Surfaces

TL;DR

This paper investigates the properties of the Laplace normal vector field on relatively normalized skew ruled surfaces in three-dimensional space, identifying conditions for degeneracy and special configurations.

Contribution

It characterizes all ruled surfaces and relative normalizations where the Laplace normal image degenerates into a point or a curve, and studies specific cases with normals on the asymptotic plane.

Findings

Identified conditions for the Laplace normal image to degenerate into a point or a curve.

Characterized ruled surfaces with normals on the asymptotic plane.

Analyzed the Laplace normal image of non-conoidal 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.