曲率和航向
曲线的曲率（curvature）就是针对曲线上某个点的切线方向角对弧长的转动率，通过微分来定义，表明曲线偏离直线的程度。数学上表明曲线在某一点的弯曲程度的数值。曲率越大，表示曲线的弯曲程度越大。曲率的倒数就是曲率半径。
设曲线直角坐标方程$y = f(x)$且具有二阶导数，那么曲率公式为： \(\begin{aligned} K = \dfrac{|y^{''}|}{(1+y^{'2})^{\dfrac{3}{2}} } \end{aligned}\) 航向为： \(\begin{aligned} Yaw = \arctan(y') \end{aligned}\)
如果曲线是由参数方程 \(\begin{aligned} x &= \psi(t) \\ y &= \omega(t) \end{aligned}\) 那么曲率为: \(\begin{aligned} K = \dfrac{|\psi'(t)\omega''(t) - \omega'(t)\psi''(t)|}{[\psi^{'2}(t) + \omega^{'2}(t)]^{\dfrac{3}{2}}} \end{aligned}\) 那么航向为: \(\begin{aligned} Yaw = \arctan(\dfrac{\omega'(t)}{\psi'(t)}) \end{aligned}\) 附录求导公式： \(\begin{aligned} \dfrac{dy}{dx} &= \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}} = \dfrac{\omega'(t)}{\psi'(t)} \\ \dfrac{d^2y}{dx^2} &= \dfrac{d}{dt}(\dfrac{dy}{dx})(\dfrac{1}{\dfrac{dx}{dt}}) = (\dfrac{dy}{dx})'(\dfrac{1}{\dfrac{dx}{dt}}) \\ &= (\dfrac{\omega'(t)}{\psi'(t)})'(\dfrac{1}{\psi'(t)}) \\ &= \dfrac{\omega^{'2}(t)\psi^{'}(t) - \psi^{'2}(t)\omega^{'}(t)}{\psi^{'3}(t)} \end{aligned}\)