Return of the plane evolute

TL;DR

This paper investigates the properties of evolutes of plane real-algebraic curves, establishing lower bounds for various intersection and singularity counts related to these evolutes for curves of degree d.

Contribution

It provides new lower bounds for intersection numbers and singularities of evolutes of real-algebraic curves of degree d, advancing understanding of their geometric properties.

Findings

Lower bounds for real line intersections with evolutes

Maximum number of real cusps on evolutes

Maximum number of nodes on dual curves and evolutes

Abstract

Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree $d\ge 2$, we provide lower bounds for the following four numerical invariants: 1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree $d$; 2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree $d$; 3) the maximal number of (cru)nodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree $d$; 4) the maximal number of (cru)nodes which can occur on the evolute of a real-algebraic curve of degree $d$.