The behaviour of curvature functions at cusps and inflection points

At a 3/2-cusp of a given plane curve $γ(t)$, both of the Euclidean curvature $κ_g$ and the affine curvature $κ_A$ diverge. In this paper, we show that each of $\sqrt{|s_g|}κ_g$ and $(s_A)^2 κ_A$ (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 $κ_g$ and $κ_A$ 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.

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