曲率

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研究曲线的弯曲程度

弧微分

\begin{aligned} Δ s & = \sqrt{(Δ x)^{2} + (Δ y)^{2}} \\ d s & = \sqrt{(d x)^{2} + (d y)^{2}} \end{aligned}

\begin{array}{r} \tan α = y^{'} \\ \sec^{2} α \frac{d α}{d x} = y^{″} \\ d α = \frac{y^{″}}{1 + y^{' 2}} d x \end{array}

单位弧段上，切线转过的角度的大小

\begin{array}{r} K = lim_{Δ x \to 0} \bar{K} = lim_{Δ x \to 0} | \frac{Δ α}{Δ s} | = | \frac{d α}{d s} | \end{array}

$d s = \sqrt{1 + y^{' 2}} d x$

\begin{array}{r} K = | \frac{\frac{y^{″}}{1 + y^{' 2}} d x}{\sqrt{1 + y^{' 2}} d x} | = \frac{| y^{″} |}{(1 + y^{' 2})^{3 / 2}} \end{array}

当 $| y^{'} | ≪ 1$ 时， $K \approx | y^{″} |$ ，曲率半径 $ρ = \frac{1}{K}$