The behavior of curvature functions at cusps and inflection points

At a 3/2-cusp of a given plane curve γ(t)γ(t), both of the Euclidean curvature κgκg and the affine curvature κAκA diverge. In this paper, we show that each of |sg|κg and (sA)2κA(sA)2κA (called the Euclidean and affine normalized curvature  , respectively) at a 3/2-cusp is a C∞C∞-function of the variable t  , where sgsg (resp. sAsA) is the Euclidean (resp. affine) arclength parameter of the curve corresponding to the 3/2-cusp sg=0sg=0 (resp. sA=0sA=0). Moreover, we give a characterization of the behavior of the curvature functions κgκg and κAκA 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.