Curvature 研究曲线的弯曲程度
曲率定义
曲率定义为：单位弧段上，切线转过的角度的大小
K=\lim\limits_{ \Delta x \to 0 } \bar{K}=\lim\limits_{ \Delta x \to 0 } \left\lvert \dfrac{\Delta \alpha}{\Delta s} \right\rvert=\left\lvert \dfrac{\mathrm{d}\alpha}{ \mathrm{d}s} \right\rvert \end{align}$$ 弧微分 $\mathrm{d}s$ $$\begin{align} \Delta s =\sqrt{ (\Delta x)^{2}+(\Delta y)^{2} } \quad \mathrm{d}s & =\sqrt{ ( \mathrm{d}x)^{2} +(\mathrm{d}y)^{2}} \quad \mathrm{d} s=\sqrt{ 1+y'^{2} }\; \mathrm{d}x \end{align}$$ $$\begin{align} \tan \alpha=y' \quad \sec ^{2}\alpha \; \dfrac{\mathrm{d}\alpha}{\; \mathrm{d}x}=y'' \quad \mathrm{d}\alpha= \dfrac{y''}{1+y'^{2}} \mathrm{d}x \end{align}$$ $$\begin{align} K= \left\lvert \dfrac{\dfrac{y''}{1+y'^{2}} \mathrm{d}x}{\sqrt{ 1+y'^{2} }\mathrm{d}x} \right\rvert= \dfrac{\left\lvert y'' \right\rvert}{(1+y'^{2})^{3 / 2}} \end{align}$$ 当 $\left\lvert y' \right\rvert\ll 1$ 时，$K \approx \left\lvert y'' \right\rvert$，曲率半径 $\rho= \dfrac{1}{K}$