1102.4478Feb 22, 201118pages, 3 figures

The behaviour of curvature functions at cusps and inflection points

Shohei Shiba, Masaaki Umehara

Abstract

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 $\sqrt{|s_g|}κ_g$

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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.