# Normal form of swallowtail and its applications

## TL;DR

This paper constructs a canonical form of the swallowtail singularity in three-dimensional space using coordinate transformations and applies it to classify the configurations of asymptotic and characteristic curves near the singularity.

## Contribution

It introduces a new normal form for swallowtail singularities in R^3 and applies it to classify related curve configurations.

## Key findings

- Established a coordinate transformation-based normal form for swallowtail singularities.
- Classified asymptotic and characteristic curve configurations near swallowtail singularities.
- Provided a framework for analyzing singularities in differential geometry.

## Abstract

We construct a form of swallowtail singularity in R^3 which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtail.