曲率函数在尖点和拐点的性质
The behaviour of curvature functions at cusps and inflection points
摘要：
在给定平面曲线$γ(t)$的3/2尖点处，欧几里得曲率$κ_g$和仿射曲率$κ_A$都发散。本文证明了在3/2尖点处的$\sqrt{|s_g|}κ_g$和$(s_A)^2 κ_A$（分别称为欧几里德和仿射归一化曲率）是变量$t$的光滑函数，其中$s_g$（分别称为Euclidean和仿射归一化曲率）是变量$t$的光滑函数. $s_A$）是欧几里得（分别为仿射）弧长参数的曲线对应的3/2尖$s_g=0$（分别）。$s_A=0$）。此外，我们给出了曲率函数$κ_g$和$κ_A$在3/2-尖点处的行为的特征。另一方面，拐点也是仿射几何中曲线的奇点。我们给出了仿射曲率函数在一般拐点附近的一个类似的刻画。作为应用，给出了3/2-尖点和一般拐点的仿射不变量。
英文摘要：
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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