The behaviour of curvature functions at cusps and inflection points

TL;DR

This paper investigates the behavior of curvature functions at cusps and inflection points of plane curves, introducing normalized curvature functions that are smooth and providing new invariants for these singularities.

Contribution

It introduces normalized curvature functions that remain smooth at cusps and inflection points, and characterizes their behavior, leading to new affine invariants for these singularities.

Findings

Normalized curvature functions are smooth at cusps and inflection points.

Characterization of curvature behavior at 3/2-cusps and inflection points.

Introduction of new affine invariants for singular points.

Abstract

At a 3/2-cusp of a given plane curve $\gamma(t)$, both of the Euclidean curvature $\kappa_g$ and the affine curvature $\kappa_A$ diverge. In this paper, we show that each of $\sqrt{|s_g|}\kappa_g$ and $(s_A)^2 \kappa_A$ (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a smooth function of the variable $t$, where $s_g$ (resp. $s_A$) is the Euclidean (resp. affine) arclength parameter of the curve corresponding to the 3/2-cusp $s_g=0$ (resp. $s_A=0$). Moreover, we give a characterization of the behaviour of the curvature functions $\kappa_g$ and $\kappa_A$ at 3/2-cusps. On the other hand, inflection points are also singular points of curves in affine geometry. We give a similar characterization of affine curvature functions near generic inflection points. As an application, new affine invariants of 3/2-cusps and generic inflection points are given.