Classifications of cusps appearing on plane curves

TL;DR

This paper classifies cusps on plane curves, focusing on $(n, n+1)$ cusps, their differential conditions, and their relation to evolutes, providing a complete classification for $(4, 5)$-cusps.

Contribution

It introduces new criteria for $(n, n+1)$ cusps and offers a complete classification for $(4, 5)$-cusps, linking singularities to evolutes of fronts.

Findings

Criteria for $(n, n+1)$ cusps established

Complete classification of $(4, 5)$-cusps provided

Relations between cusps and evolutes analyzed

Abstract

In this paper, we deal with plane curves with cusps. It is well known that there are various types of cusps. Among them, we investigate criteria for $(n, n+1)$ cusps with respect to several differential conditions and relations between these singularities and evolutes of fronts. We give complete classifications with respect to $(4, 5)$-cusps.