Flat surfaces along swallowtails

TL;DR

This paper studies developable flat surfaces along swallowtail singularities, introducing Darboux frames and invariants to analyze their properties, and provides a new example of a frontal resembling a swallowtail.

Contribution

It introduces Darboux frames and invariants for swallowtail singularities, offering new tools for analyzing developable surfaces along these singularities.

Findings

Defined Darboux frames along swallowtails

Introduced invariants for singularity analysis

Constructed a new example of a swallowtail-like frontal

Abstract

We consider developable surfaces along the singular set of a swallowtail which are considered to be flat approximations of the swallowtail. For the study of singularities of such developable surfaces, we introduce the notion of Darboux frames along swallowtails and invariants. As a by-product, we give a new example of a frontal which is locally homeomorphic to a swallowtail.