Recovering plane curves of low degree from their inflection lines and inflection points

TL;DR

This paper investigates whether smooth plane curves of degree three and four can be uniquely reconstructed from their inflection lines and points, providing positive results under certain conditions.

Contribution

It demonstrates that a general smooth plane quartic can be recovered from its inflection lines and a single inflection point, and that all smooth plane cubics can be reconstructed from their inflection data.

Findings

Quartic curves can be recovered from inflection lines and a point.

Cubic curves can be recovered from inflection lines alone.

Positive reconstruction results for specific degrees.

Abstract

In this paper we consider the following problem: is it possible to recover a smooth plane curve of degree at least three from its inflection lines? We answer positively to the posed question for a general smooth plane quartic curve, making the additional assumption that also one inflection point is given, and for any smooth plane cubic curve.