A property that characterizes the Enneper surface and helix surfaces

TL;DR

This paper characterizes Enneper and helix surfaces in 3D space by their isogonal lines, showing they are unique in having these lines as generalized helices and pseudo-geodesics.

Contribution

It establishes a unique geometric property that distinguishes Enneper and helix surfaces based on their isogonal lines in Euclidean space.

Findings

Enneper and helix surfaces are uniquely characterized by their isogonal lines.

Their isogonal lines are generalized helices and pseudo-geodesics.

The paper provides a new geometric characterization of these surfaces.

Abstract

The main goal of this paper is to show that helix surfaces and the Enneper surface are the only surfaces in the 3-dimensional Euclidean space $R^3$ whose isogonal lines are generalized helices and pseudo-geodesic lines.