曲率

式チートシート

$ R=\frac{(1+f'(x)^2)^\frac32}{|f''(x)|}

微少量で表す場合

$ R=\frac{((\mathrm{d}y)^2+(\mathrm{d}x)^2)^\frac32}{\left|\frac{\mathrm{d}^2y}{{\mathrm{d} x}^2}\right|(\mathrm{d}x)^3}

媒介変数表示

$ R=\frac{|\mathrm d(x,y)|^3}{y''\mathrm dx-x''\mathrm dy}

導出

$ \therefore|\mathrm d\bm r|=\rho|\mathrm d\hat{\dot{\bm r}}|

$ \iff\rho=\frac{|\mathrm d\bm r|}{|\mathrm d\hat{\dot{\bm r}}|}

$ =\frac{|\dot{\bm r}|}{|\dot{\hat{\dot{\bm r}}}|}

$ =\frac{|\dot{\bm r}|^2}{|(\bm I-\hat{\dot{\bm r}}\hat{\dot{\bm r}})\cdot\ddot{\bm r}|}

$ =\frac{|\dot{\bm r}|^2}{\sqrt{|\ddot{\bm r}|^2-(\hat{\dot{\bm r}}\cdot\ddot{\bm r})^2}}

函数形式の導出

2次元に限定する

$ \dot{\bm r}=r_x'(t)\bm e_x+f'(r_x(t))r_x'(t)\bm e_y=r_x'(t)(\bm e_x+f'(r_x(t))\bm e_y)

$ |\dot{\bm r}|^2={r_x'(t)}^2\left(1+{f'(r_x(t))}^2\right)

$ \ddot{\bm r}=r_x''(t)\bm e_x+f''(r_x(t)){r_x'(t)}^2\bm e_y+f'(r_x(t))r_x''(t)\bm e_y

$ |\ddot{\bm r}|^2={r_x''(t)}^2+\left(f''(r_x(t)){r_x'(t)}^2+f'(r_x(t))r_x''(t)\right)^2

$ =\frac{{r_x''(t)}^2}{{r_x'(t)}^2}|\dot{\bm r}|^2+f''(r_x(t)){r_x'(t)}^2\left(f''(r_x(t)){r_x'(t)}^2+2f'(r_x(t))r_x''(t)\right)

$ \dot{\bm r}\cdot\ddot{\bm r}=r_x'(t)r_x''(t)+f'(r_x(t))f''(r_x(t)){r_x'(t)}^3+{f'(r_x(t))}^2r_x'(t)r_x''(t)

$ =\frac{r_x''(t)}{r_x'(t)}|\dot{\bm r}|^2+f'(r_x(t))f''(r_x(t)){r_x'(t)}^3

$ |\ddot{\bm r}|^2-(\hat{\dot{\bm r}}\cdot\ddot{\bm r})^2=\frac{{r_x''(t)}^2}{{r_x'(t)}^2}|\dot{\bm r}|^2+f''(r_x(t)){r_x'(t)}^2\left(f''(r_x(t)){r_x'(t)}^2+2f'(r_x(t))r_x''(t)\right)-\left(\frac{r_x''(t)}{r_x'(t)}|\dot{\bm r}|+f'(r_x(t))f''(r_x(t)){r_x'(t)}^2\frac{r_x'(t)}{|\dot{\bm r}|}\right)^2

$ ={f''(r_x(t))}^2{r_x'(t)}^4+2f'(r_x(t))f''(r_x(t)){r_x'(t)}^2r_x''(t)

$ -f'(r_x(t))f''(r_x(t)){r_x'(t)}^2\frac{r_x'(t)}{|\dot{\bm r}|}\left(2\frac{r_x''(t)}{r_x'(t)}|\dot{\bm r}|+f'(r_x(t))f''(r_x(t)){r_x'(t)}^2\frac{r_x'(t)}{|\dot{\bm r}|}\right)

$ ={f''(r_x(t))}^2{r_x'(t)}^4+2f'(r_x(t))f''(r_x(t)){r_x'(t)}^2r_x''(t)

$ -2f'(r_x(t))f''(r_x(t)){r'_x(t)}^2r_x''(t)-{f'(r_x(t))}^2{f''(r_x(t))}^2\frac{{r_x'(t)}^6}{|\dot{\bm r}|^2}

$ ={f''(r_x(t))}^2{r_x'(t)}^4-{f'(r_x(t))}^2{f''(r_x(t))}^2\frac{{r_x'(t)}^6}{|\dot{\bm r}|^2}

$ ={f''(r_x(t))}^2{r_x'(t)}^4\left(1-{f'(r_x(t))}^2\frac{{r_x'(t)}^2}{|\dot{\bm r}|^2}\right)

$ ={f''(r_x(t))}^2{r_x'(t)}^4\left(1-\frac{|\dot{\bm r}|^2-{r_x'(t)}^2}{|\dot{\bm r}|^2}\right)

$ ={f''(r_x(t))}^2{r_x'(t)}^4\frac{{r_x'(t)}^2}{|\dot{\bm r}|^2}

$ =\left(\frac{f''(r_x(t))}{|\dot{\bm r}|}{r_x'(t)}^3\right)^2

$ \therefore\rho=\frac{|\dot{\bm r}|^2}{\sqrt{|\ddot{\bm r}|^2-(\hat{\dot{\bm r}}\cdot\ddot{\bm r})^2}}

$ =\frac{|\dot{\bm r}|^2}{\frac{f''(r_x(t))}{|\dot{\bm r}|}{r_x'(t)}^3}

$ =\frac{\left(\frac{|\dot{\bm r}|}{r_x'(t)}\right)^3}{f''(r_x(t))}

$ =\frac{\left(1+{f'(r_x(t))}^2\right)^\frac32}{f''(r_x(t))}

$ g(x):=(1+x)^\alpha

$ g(x^2)'=2g'(x^2)x

$ g(x^2)''=4g''(x^2)x^2+2g'(x^2)

$ g(x^2)'''=8g'''(x^2)x^3+8g''(x^2)x+4g''(x^2)x

$ =8g'''(x^2)x^3+12g''(x^2)x

$ g(x^2)''''=16g''''(x^2)x^4+24g'''(x^2)x^2+24g'''(x^2)x^2+12g''(x^2)

$ =16g''''(x^2)x^4+48g'''(x^2)x^2+12g''(x^2)

$ \therefore(1+x^2)^\alpha=1+0+\frac12\cdot2g'(0)x^2+0+\frac1{4!}\cdot12g''(0)x^4+\cdots

$ =1+\alpha(1+0)^{\alpha-1}x^2+\frac12\alpha^{\underline2}(1+0)^{\alpha-2}x^4+\cdots

$ =1+\alpha x^2+\frac12\alpha^{\underline2}x^4+\cdots

$ \therefore(1+x^2)^\frac32=1+\frac32x^2+\frac38x^4+\cdots

$ \pmb c=\pmb c_0+R\begin{pmatrix}\cos\theta\\\sin\theta\end{pmatrix}

$ \dot{\pmb l}(t)//\dot{\pmb c}

$ \iff \dot l_x=-Rk\sin\theta\land\dot l_y=Rk\cos\theta\quad\text{.for }\exists k\in\R

$ \ddot{\pmb l}(t)=\ddot{\pmb c}

$ \iff \ddot l_x=-R\cos\theta\land\ddot l_y=-R\sin\theta

$ \mathrm{d}l_x=-\tan\theta{\mathrm{d}l_y}

$ \implies \mathrm{d}^2l_x=-\frac{\mathrm{d}\theta}{(\cos\theta)^2}\mathrm{d}l_y-\tan\theta\mathrm{d}^2l_y=-(1+(\tan\theta)^2)\mathrm{d}\theta\mathrm{d}l_y-\tan\theta\mathrm{d}^2l_y

$ \iff \mathrm{d}\theta=-\frac{\mathrm{d}^2l_x+\tan\theta\mathrm{d}^2l_y}{1+(\tan\theta)^2}=-(\cos\theta)^2\mathrm d^2l_x+\cos\theta\sin\theta\mathrm d^2l_y

$ \begin{dcases}\mathrm{d}\dot l_x=-Rk\cos\theta\mathrm{d}\theta=-\dot l_y\mathrm{d}\theta\\\mathrm{d}\dot l_y=-Rk\sin\theta\mathrm{d}\theta\end{dcases}

$ \iff

$ |\mathrm d\pmb l|^2=R^2k^2(\mathrm d\theta)^2

$ \mathrm d\dot f=\mathrm d\frac{\mathrm d f}{\mathrm d t}=\frac{\mathrm d^2f\mathrm dt-\mathrm d^2t\mathrm df}{\mathrm dt}=\mathrm d^2f-(\mathrm dt)'\mathrm df

曲線の曲がり具合を示す量．

曲率が大きいほど曲がり具合はきつい．

曲線の一階微分は接線の傾きであり，その接線の傾きがどう変化していくかを表したものが曲率なので，曲線の二階微分で表される．

実際にはかなり複雑な非線型函数になる

半径 R の円の曲率は 1/R．