曲率

定义

在曲线上由点 $M$ 到点 $M'$: 切线转过角度 $|\Delta \alpha|$, 弧长为 $\Delta s$ 则平均曲率: $\dfrac{|\Delta \alpha|}{|\Delta s|}$曲率: $K(M) = \lim_{|\Delta s| \to 0} \dfrac{|\Delta \alpha|}{|\Delta s|}$

注意

注意曲率要加绝对值，曲率是非负数

直角坐标系

由于 $y' = \tan \alpha$, $\alpha = \arctan y'$$\implies d\alpha = \dfrac{y''}{1+y'^2} dx$ 又因为 $ds = \sqrt{1+y'^2} dx$

$$\implies K = \left|\dfrac{d\alpha}{ds}\right| = \left|\dfrac{y''}{(1+y'^2)^\frac{3}{2}}\right|$$

警告

注意曲率为非负数，要加绝对值

参数方程

见 高阶导数#参数方程

$$\dfrac{dy}{dx} = \dfrac{y'(t)}{x'(t)}$$

$$\dfrac{d^2 y}{dx^2} = \dfrac{d}{dx}(\dfrac{dy}{dx}) = \dfrac{\dfrac{d}{dt}(\dfrac{dy}{dx})}{\dfrac{dx}{dt}} = \dfrac{x'(t) y''(t) - x''(t) y'(t)}{x'^3(t)}$$

故

$$K = \dfrac{|x'(t) y''(t) - x''(t) y'(t)|}{[x'^2(t) + y'^2(t)]^\frac{3}{2}}$$

曲率圆

曲率半径: $R = \dfrac{1}{K}$ (曲率的倒数) 曲率中心坐标:

$$\left( x-\dfrac{y'(1+y'^2)}{y''}, y+\dfrac{1+y'^2}{y''} \right)$$