Whitney umbrellas and swallowtails

TL;DR

This paper establishes a mathematical correspondence between Whitney umbrellas of pedal unfolding type and normalized swallowtails using concepts from Legendrian singularity theory, revealing a deep geometric connection.

Contribution

It introduces new notions of pedal unfolding type and normalized Legendrian map-germs, and demonstrates a natural one-to-one correspondence between Whitney umbrellas and swallowtails.

Findings

Established a correspondence between Whitney umbrellas and swallowtails.

Introduced the concepts of pedal unfolding type and normalized Legendrian map-germs.

Connected the fundamental theorem of calculus to singularity theory.

Abstract

In this paper, we introduce the notions of map-germs of pedal unfolding type and normalized Legendrian map-germs; and then we show that the fundamental theorem of calculus provides a natural one to one correspondence between Whitney umbrellas of pedal unfolding type and normalized swallowtails.