曲率：曲线的弯曲程度。

一个圆半径$r$越小，弯曲程度越大，于是定义曲率$k$。

$$k=\frac{1}{r}$$

对于一段圆弧$s$，对应半径$r$和角度$\alpha$，则有

$$s=\alpha r \rightarrow r = \frac{s}{\alpha}\rightarrow \\ k = \frac{1}{r}=\frac{\alpha}{s}$$

对于一般曲线在某一点上的曲率，可以取其微分

$$\begin{equation}k=\left| \frac{d\alpha}{ds} \right|\end{equation}$$

设函数$f(x)$二阶导数存在。有

$$\frac{dy}{dx}=\tan \alpha \text{，导数定义为切线斜率}$$ $$\begin{align*} y'' & = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{d\alpha} \frac{d\alpha}{dx} \cdot \tan\alpha = \frac{d\alpha}{dx} \cdot \sec^{2}\alpha \\ \Rightarrow \frac{d\alpha}{dx} & = \frac{y''}{\sec^{2}\alpha} = \frac{y''}{1 + \tan^{2}\alpha} = \frac{y''}{1 + y'^{2}} \\ \Rightarrow d\alpha & = \frac{y''}{1 + y'^{2}} \, dx \end{align*}$$

弧微分为

$$ds = \sqrt{1 + y'^{2}} \, dx \quad % 修正原图笔误（添加 y' 的导数符号）$$

代入到曲率定义，

$$\begin{equation}k = \left| \frac{d \alpha}{ds} \right| = \frac{|y''|}{(1 + y'^2)^\frac{3}{2}} \text{ (2)}\end{equation}$$

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如果在参数方程下呢？

对于曲线参数方程

$$\begin{cases} x = \varphi(t) \\ y = \psi(t) \quad % 修正：原图 \gamma 应为 \psi 保持一致性 \end{cases}$$

一阶导数

$$y' = \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\psi'(t)}{\varphi'(t)} \\[0.8em]$$

二阶导数

$$y'' = \frac{d}{dx}\left( \frac{dy}{dx} \right) = \frac{d}{dt}\left( \frac{\psi'(t)}{\varphi'(t)} \right) \cdot \frac{dt}{dx} \\ = \left[ \frac{\psi''(t)\varphi'(t) - \psi'(t)\varphi''(t)}{[\varphi'(t)]^2} \right] \cdot \frac{1}{\varphi'(t)} \quad % 修正分母指数 = \frac{\psi''(t)\varphi'(t) - \psi'(t)\varphi''(t)}{[\varphi'(t)]^3}$$

代入到(2)，

$$k = \left| \frac{y''}{(1 + y'^2)^{3/2}} \right| = \frac{ |\psi''(t)\varphi'(t) - \psi'(t)\varphi''(t)| / |\varphi'(t)|^3 }{ \left( 1 + \left[ \frac{\psi'(t)}{\varphi'(t)} \right]^2 \right)^{3/2} } \\ = \frac{ |\psi''(t)\varphi'(t) - \psi'(t)\varphi''(t)| }{ \left( [\varphi'(t)]^2 + [\psi'(t)]^2 \right)^\frac{3}{2} }$$

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