The Hesse pencil of plane curves and osculating conics

TL;DR

This paper investigates osculating conics and sextactic points on algebraic curves, focusing on the Hesse pencil of plane cubics, deriving explicit formulas, and constructing new free and nearly free curves.

Contribution

It provides explicit coordinate formulas for special points on Hesse pencil curves and introduces new families of free and nearly free curves.

Findings

Explicit formulas for osculating conics and sextactic points.

Construction of new free and nearly free curves.

Clarification of classical geometric approaches.

Abstract

In this paper, we revisit the classical problem of determining osculating conics and sextactic points for a given algebraic curve. Our focus is on a particular family of plane cubic curves known as the Hesse pencil. By employing classical tools from projective differential geometry, we derive explicit coordinates for these special points. The resulting formulas not only clarify previous approaches but also lead to the construction of new families of free and nearly free curves, extending recent findings the freeness of curves.