Generic singularities of line fields on 2D manifolds

TL;DR

This paper introduces a new classification approach for generic singularities of line fields on 2D manifolds by representing them as bisectors of vector field pairs, revealing their topological types.

Contribution

It proposes a novel method to classify line field singularities using vector field pairs and conformal structures, extending previous studies beyond principal curvature lines.

Findings

Singularities are topologically equivalent to Lemon, Star, and Monstar types.

Line fields can be characterized as bisectors of vector field pairs.

The approach applies to general line fields, not just curvature lines.

Abstract

Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with respect to a given conformal structure. The singularities correspond to the zeros of the vector fields and the genericity is considered with respect to a natural topology in the space of pairs of vector fields. Line fields at generic singularities turn out to be topologically equivalent to the Lemon, Star and Monstar singularities that one finds at umbilical points.