# 曲率

$\kappa =\underset{\Delta s\to 0}{\mathrm{lim}}\frac{\Delta \theta }{\Delta s}$

と定義される．すなわち，曲率 $\kappa$ は曲線上の点の移動距離に関する進行移動方向の変化率である．

$\kappa =\frac{{f}^{″}\left({x}_{\text{A}}\right)}{{\left[1+{\left\{{f}^{\prime }\left({x}_{\text{A}}\right)\right\}}^{2}\right]}^{\frac{3}{2}}}$

となる．

## 導出

$\Delta \theta ={\theta }_{\text{B}}-{\theta }_{\text{A}}$

$\mathrm{tan}\Delta \theta =\mathrm{tan}\left({\theta }_{B}-{\theta }_{A}\right)$

$=\frac{\mathrm{tan}{\theta }_{\mathrm{\text{B}}}-\mathrm{tan}{\theta }_{\mathrm{\text{A}}}}{1+\mathrm{tan}{\theta }_{\mathrm{\text{B}}}\mathrm{tan}{\theta }_{\mathrm{\text{A}}}}$

$\mathrm{tan}{\theta }_{\text{A}}={f}^{\prime }\left({x}_{\text{A}}\right),\mathrm{tan}{\theta }_{\text{B}}={f}^{\prime }\left({x}_{\text{B}}\right)$ より

$=\frac{{f}^{\prime }\left({x}_{\text{B}}\right)-{f}^{\prime }\left({x}_{\text{A}}\right)}{1+{f}^{\prime }\left({x}_{\text{B}}\right){f}^{\prime }\left({x}_{\text{A}}\right)}$

$\Delta \theta ={\mathrm{tan}}^{-1}\frac{{f}^{\prime }\left({x}_{\text{B}}\right)-{f}^{\prime }\left({x}_{\text{A}}\right)}{1+{f}^{\prime }\left({x}_{\text{B}}\right){f}^{\prime }\left({x}_{\text{A}}\right)}$

${\mathrm{tan}}^{-1}x$ のマクローリン展開 ${\mathrm{tan}}^{-1}x=x-\frac{1}{3}{x}^{3}+\frac{1}{5}{x}^{5}-\cdots$ より

$=\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{B}\right){f}^{\prime }\left({x}_{A}\right)}-\frac{1}{3}{\left(\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{B}\right){f}^{\prime }\left({x}_{A}\right)}\right)}^{3}+\frac{1}{5}{\left(\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{B}\right){f}^{\prime }\left({x}_{A}\right)}\right)}^{5}-\cdots$

$f\left({x}_{C}\right)=\frac{1}{{x}_{B}-{x}_{A}}\underset{{x}_{A}}{\overset{{x}_{B}}{\int }}f\left(x\right)dx$

$\sqrt{1+{\left\{{f}^{\prime }\left({x}_{C}\right)\right\}}^{2}}=\frac{1}{{x}_{B}-{x}_{A}}\underset{{x}_{A}}{\overset{{x}_{B}}{\int }}\sqrt{1+{\left\{{f}^{\prime }\left(x\right)\right\}}^{2}}dx$

が成り立つ

よって

$\Delta s=\underset{{x}_{\text{A}}}{\overset{{x}_{\text{B}}}{\int }}\sqrt{1+{\left\{f{}^{\prime }\left(x\right)\right\}}^{2}}\mathrm{dx}$ $=\sqrt{1+{\left\{f{}^{\prime }\left({x}_{\text{C}}\right)\right\}}^{2}}\left({x}_{\text{B}}-{x}_{\text{A}}\right)$

$\Delta s\to 0$ のとき ${x}_{\text{B}}\to {x}_{\text{A}}$ となる。

よって曲率Kは,

$K=\underset{\Delta s\to 0}{\mathrm{lim}}\frac{\Delta \theta }{\Delta s}$

$=\underset{{x}_{B}\to {x}_{A}}{\mathrm{lim}}\frac{1}{\sqrt{1+{\left\{{f}^{\prime }\left({x}_{C}\right)\right\}}^{2}}\left({x}_{B}-{x}_{A}\right)}\left\{\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{A}\right){f}^{\prime }\left({x}_{B}\right)}-\frac{1}{3}{\left(\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{A}\right){f}^{\prime }\left({x}_{B}\right)}\right)}^{3}+\frac{1}{5}{\left(\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{A}\right){f}^{\prime }\left({x}_{B}\right)}\right)}^{5}-\cdots \right\}$

$=\underset{{x}_{B}\to {x}_{A}}{\mathrm{lim}}\frac{1}{\sqrt{1+{\left\{{f}^{\prime }\left({x}_{C}\right)\right\}}^{2}}\left({x}_{B}-{x}_{A}\right)}\cdot \frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{A}\right){f}^{\prime }\left({x}_{B}\right)}\left\{1-\frac{1}{3}{\left(\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{A}\right){f}^{\prime }\left({x}_{B}\right)}\right)}^{2}+\frac{1}{5}{\left(\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{A}\right){f}^{\prime }\left({x}_{B}\right)}\right)}^{4}-\cdots \right\}$

${x}_{B}\to {x}_{A}$ のとき${x}_{C}\to {x}_{A}$ また,導関数の定義より

$\underset{{x}_{B}\to {x}_{A}}{\mathrm{lim}}\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{1+{f}^{\prime }\left({x}_{A}\right){f}^{\prime }\left({x}_{B}\right)}=0$

$\underset{{x}_{B}\to {x}_{A}}{\mathrm{lim}}\frac{{f}^{\prime }\left({x}_{B}\right)-{f}^{\prime }\left({x}_{A}\right)}{{x}_{B}-{x}_{A}}={f}^{″}\left({x}_{A}\right)$

が成り立つ

よって

$k=\frac{1}{\sqrt{1+{\left\{{f}^{\prime }\left({x}_{A}\right)\right\}}^{2}}}\cdot \frac{1}{1+{f}^{\prime }\left({x}_{A}\right)\cdot {f}^{\prime }\left({x}_{A}\right)}\cdot {f}^{″}\left({x}_{A}\right)$

$=\frac{{f}^{″}\left({x}_{A}\right)}{{\left[\sqrt{1+{\left\{{f}^{\prime }\left({x}_{A}\right)\right\}}^{2}}\right]}^{\frac{3}{2}}}$

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