陰関数表示された曲線の曲率半径

$3$\displaystyle{xy}$ 平面で定義された曲線が陰関数表示 $3$\displaystyle{ F$x,y$=0 }$ で表されているとし，その偏導関数を

$3$\displaystyle{ \hspace{1}{F}_{x}=\frac{\hspace{2} {\partial}F \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ ， $3$\displaystyle{ \hspace{1}{F}_{y}=\frac{\hspace{2} {\partial}F \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ ， $3$\displaystyle{ \hspace{1}{F}_{ xx }=\frac{\hspace{2} {{\partial}}^{2}F \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$ ， $3$\displaystyle{ \hspace{1}{F}_{ xy }=\frac{\hspace{2} {{\partial}}^{2}F \hspace{2}}{\hspace{2} {\partial}y{\partial}x \hspace{2}} }$ ， $3$\displaystyle{ \hspace{1}{F}_{ yx }=\frac{\hspace{2} {{\partial}}^{2}F \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}} }$ ， $3$\displaystyle{ \hspace{1}{F}_{ yy }=\frac{\hspace{2} {{\partial}}^{2}F \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$

とする． $3$\displaystyle{ F$x,y$=0 }$ より，全微分 $3$\displaystyle{ dF=\hspace{1}{F}_{x}dx+\hspace{1}{F}_{y}dy=0 }$ なので

$3$\displaystyle{ \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}}=- \frac{\hspace{2} \hspace{1}{F}_{x} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}} }$

である．曲率半径

$3$\displaystyle{ R= \| \frac{\hspace{2} ds \hspace{2}}{\hspace{2} d\alpha \hspace{2}} \| }$

において

$3$\displaystyle{ \tan \alpha =\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}}=- \frac{\hspace{2} \hspace{1}{F}_{x} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}} }$

より

$3$\displaystyle{ \frac{\hspace{2} d$\tan \alpha $ \hspace{2}}{\hspace{2} d\alpha \hspace{2}}d\alpha =- d $ \frac{\hspace{2} \hspace{1}{F}_{x} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}} $ }$ $3$\displaystyle{ =- \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \frac{\hspace{2} \hspace{1}{F}_{x} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}} $ dx- \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ \frac{\hspace{2} \hspace{1}{F}_{x} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}} $ \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}}dx }$

⇒ $3$\displaystyle{ \frac{\hspace{2} d\alpha \hspace{2}}{\hspace{2} { \cos }^{2}\alpha \hspace{2}}=- $\frac{\hspace{2} \hspace{1}{F}_{ xx }\hspace{1}{F}_{y}- \hspace{1}{F}_{x}\hspace{1}{F}_{ yx } \hspace{2}}{\hspace{2} F_{y}^{2} \hspace{2}}$ dx- $\frac{\hspace{2} \hspace{1}{F}_{ xy }\hspace{1}{F}_{y}- \hspace{1}{F}_{x}\hspace{1}{F}_{ yy } \hspace{2}}{\hspace{2} F_{y}^{2} \hspace{2}}$ $- \frac{\hspace{2} \hspace{1}{F}_{x} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}}$ dx }$

⇒ $3$\displaystyle{ $1+{ \tan }^{2}\alpha $ d\alpha =\frac{\hspace{2} \hspace{1}{F}_{y} $\hspace{1}{F}_{x}\hspace{1}{F}_{ yx }- \hspace{1}{F}_{y}\hspace{1}{F}_{ xx }$ - \hspace{1}{F}_{x} $\hspace{1}{F}_{x}\hspace{1}{F}_{ yy }- \hspace{1}{F}_{y}\hspace{1}{F}_{ xy }$ \hspace{2}}{\hspace{2} F_{y}^{3} \hspace{2}}dx }$

となり，右辺を行列式を用いて表すと

⇒ $3$\displaystyle{ \{1+{ $- \frac{\hspace{2} \hspace{1}{F}_{x} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}}$ }^{2}\} d\alpha }$ $3$\displaystyle{ =\frac{\hspace{2} - \hspace{1}{F}_{x} \|\begin{array} \hspace{1}{F}_{x} & \hspace{1}{F}_{ xy } & \vspace{6}\\ \hspace{1}{F}_{y} & \hspace{1}{F}_{ yy } & \vspace{6}\\ \end{array}\| +\hspace{1}{F}_{y} \|\begin{array} \hspace{1}{F}_{x} & \hspace{1}{F}_{ xx } & \vspace{6}\\ \hspace{1}{F}_{y} & \hspace{1}{F}_{ yx } & \vspace{6}\\ \end{array}\| \hspace{2}}{\hspace{2} F_{y}^{3} \hspace{2}}dx }$

⇒ $3$\displaystyle{ \frac{\hspace{2} F_{x}^{2}+F_{y}^{2} \hspace{2}}{\hspace{2} F_{y}^{2} \hspace{2}}d\alpha =\frac{\hspace{2} \text{\hspace{1}}\|\begin{array}0&\hspace{1}{F}_{x}&\hspace{1}{F}_{y}& \vspace{6}\\ \hspace{1}{F}_{x}&\hspace{1}{F}_{ xx }&\hspace{1}{F}_{ xy }& \vspace{6}\\ \hspace{1}{F}_{y}&\hspace{1}{F}_{ yx }&\hspace{1}{F}_{ yy }& \vspace{6}\\ \end{array}\|\text{\hspace{1}} \hspace{2}}{\hspace{2} F_{y}^{3} \hspace{2}}dx }$ $3$\displaystyle{ =\frac{\hspace{2} \det M \hspace{2}}{\hspace{2} F_{y}^{3} \hspace{2}}dx }$ ， $3$\displaystyle{ M= $\begin{array}0&\hspace{1}{F}_{x}&\hspace{1}{F}_{y}& \vspace{6}\\ \hspace{1}{F}_{x}&\hspace{1}{F}_{ xx }&\hspace{1}{F}_{ xy }& \vspace{6}\\ \hspace{1}{F}_{y}&\hspace{1}{F}_{ yx }&\hspace{1}{F}_{ yy }& \vspace{6}\\ \end{array}$ }$

を得る．最終的に

$3$\displaystyle{ d\alpha =\frac{\hspace{2} \det M \hspace{2}}{\hspace{2} \hspace{1}{F}_{y}$F_{x}^{2}+F_{y}^{2}$ \hspace{2}}dx }$

を得る．また

$3$\displaystyle{ ds=\sqrt{ { $dx$ }^{2}+{ $dy$ }^{2} } }$ $3$\displaystyle{ =\sqrt{ 1+{ $ \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}} $ }^{2} }dx }$ $3$\displaystyle{ =\sqrt{ 1+\frac{\hspace{2} F_{y}^{2} \hspace{2}}{\hspace{2} F_{x}^{2} \hspace{2}} }dx }$ $3$\displaystyle{ =\frac{\hspace{2} \sqrt{F_{x}^{2}+F_{y}^{2}} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}}dx }$

であるので，曲率半径 $3$\displaystyle{R}$ は $3$\displaystyle{x}$ と $3$\displaystyle{y}$ の関数として

$3$\displaystyle{ R$x,y$= \| \frac{\hspace{2} ds \hspace{2}}{\hspace{2} d\alpha \hspace{2}} \| }$ $3$\displaystyle{ = \| \frac{\hspace{2} \sqrt{F_{x}^{2}+F_{y}^{2}} \hspace{2}}{\hspace{2} \hspace{1}{F}_{y} \hspace{2}}\cdot \frac{\hspace{2} \hspace{1}{F}_{y}$F_{x}^{2}+F_{y}^{2}$ \hspace{2}}{\hspace{2} \det M \hspace{2}} \| }$ $3$\displaystyle{ =\frac{\hspace{2} { $F_{x}^{2}+F_{y}^{2}$ }^{ \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} } \hspace{2}}{\hspace{2} \|\det M\| \hspace{2}} }$

と求まる．

また，曲線上の点 P $3$\displaystyle{ $x,y$ }$ 付近を近似する円の中心 C の座標 $3$\displaystyle{ $\hspace{1}{c}_{x},\hspace{1}{c}_{y}$ }$ は

$3$\displaystyle{ $\hspace{1}{c}_{x}\text{\hspace{1}},\text{\hspace{1}}\hspace{1}{c}_{y}$=$x\text{\hspace{1}},\text{\hspace{1}}y$+ $ - \frac{\hspace{2} dy \hspace{2}}{\hspace{2} d\alpha \hspace{2}}\text{\hspace{1}},\text{\hspace{1}}\frac{\hspace{2} dx \hspace{2}}{\hspace{2} d\alpha \hspace{2}} $ }$ $3$\displaystyle{ =$x\text{\hspace{1}},\text{\hspace{1}}y$+\frac{\hspace{2} F_{x}^{2}+F_{y}^{2} \hspace{2}}{\hspace{2} \det M \hspace{2}} $ \hspace{1}{F}_{x}\text{\hspace{1}},\text{\hspace{1}}\hspace{1}{F}_{y} $ }$

となる．

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最終更新日：2023年10月1日