Geometric deformations of implicit curves

TL;DR

This paper classifies how implicit plane curves with singularities deform, focusing on the emergence of inflections and vertices during such deformations, especially in generic two-parameter families.

Contribution

It provides a classification of deformations of implicit singular plane curves concerning inflections and vertices for codimension ≤ 2.

Findings

Classified deformations with respect to inflections and vertices

Focused on generic cases in 2-parameter families

Identified conditions for emergence of geometric features

Abstract

Let f = 0 be an implicit singular plane curve. When deforming f = 0, inflections and vertex emerge from the singularities. In this papper, we classify the deformations of f = 0 with respect to the inflections and the vertices in the cases of codimension less than or equal to 2, that is, in the cases that occur generically in families of implicit curves with 2 parameters.