曲率公式、参数方程确定的曲线公式和曲率半径的推导

曲率公式、参数方程确定的曲线公式和曲率半径的推导

前言：重在记录，可能出错。

一、总结

曲率： $$\color{red} {K=\frac{{ \left| {{y '' }} \right| }}{{{ \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}}\right. }\mathop{ {\left) \right.} }\nolimits^{{\frac{{3}}{{2}}}}}}}$$

参数方程确定的曲线的曲率： $$\color{red}{K=\frac{{ \left| {{{ \omega '' }} \left( t \left) { \varphi \prime } \left( t \left) -{ \omega \prime } \left( t \left) {{ \varphi '' }} \left( t \left) \right. \right. \right. \right. \right. \right. \right. \right. } \right| }}{{ \left[ {{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }+{{ \omega \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }} \left] \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}}$$

曲率半径： $$\color{red}{{ \rho =\frac{{1}}{{K}}=}\frac{{ \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}} \left) \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}{{ \left| {{y '' }} \right| }}}$$

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二、推导

  曲线就是针对曲线上某个点的切线方向角对弧长的转动率。

  曲率越大，表示曲线的弯曲程度越大。

  曲率用K表示。切线方向角用α表示。弧长用s表示。

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1.曲率公式的推导

$${\begin{array}{*{20}{l}}{{K=\mathop{{lim}}\limits_{{ \Delta s \to 0}}{ \left| {\frac{{ \Delta \alpha }}{{ \Delta s}}} \right| }=}{ \left| {\frac{{d \alpha }}{{ds}}} \right| }={ \left| {\frac{{d \alpha }}{{dx}}∙\frac{{1}}{{\frac{{ds}}{{dx}}}}} \right| }}\\{}\\{\text{先}\text{求}\frac{{d \alpha }}{{dx}}\text{：}}\\{\frac{{dy}}{{dx}}={y \prime }=tan \alpha }\\{ \because \text{ }{y '' }=\frac{{d\mathop{{}}\nolimits^{{2}}y}}{{dx\mathop{{}}\nolimits^{{2}}}}=\frac{{d \left( \frac{{dy}}{{dx}} \right) }}{{dx}}=\frac{{d \left( \frac{{dy}}{{dx}} \right) }}{{d \alpha }}∙{\frac{{{d \alpha }}}{{dx}}}=\frac{{d \left( tan \alpha \right) }}{{d \alpha }}∙\frac{{d \alpha }}{{dx}}}\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=sec\mathop{{}}\nolimits^{{2}} \alpha {∙{\frac{{{d \alpha }}}{{dx}}}}}\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }= \left( 1+tan\mathop{{}}\nolimits^{{2}} \alpha \left) ∙\frac{{d \alpha }}{{dx}}\right. \right. }\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }= \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}} \left) \frac{{d \alpha }}{{dx}}\right. \right. }\\\color{red}{ \therefore \frac{{d \alpha }}{{dx}}=\frac{{{y '' }}}{{1+{y \prime }\mathop{{}}\nolimits^{{2}}}}}\\{}\\{}\\{\text{再}\text{求}\frac{{ds}}{{dx}}：}\\{\text{设}A\text{点}\text{坐}\text{标} \left( x\mathop{{}}\nolimits_{{1}},y\mathop{{}}\nolimits_{{1}} \left) \text{，}B\text{点}\text{坐}\text{标} \left( x\mathop{{}}\nolimits_{{2}},y\mathop{{}}\nolimits_{{2}} \right) \right. \right. }\\{\text{当}AB\text{两}\text{点}\text{无}\text{限}\text{接}\text{近}\text{时}，\text{可}\text{得} \Delta x \to 0\text{，} \Delta y \to 0\text{，}dx= \Delta x=x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}}\text{，}dy= \Delta y=y\mathop{{}}\nolimits_{{2}}-y\mathop{{}}\nolimits_{{1}}}\\{\text{弧}\text{长}\mathop{{AB}}\limits^{︵}\text{值}\text{可}\text{以}\text{用}\text{线}\text{段}AB\text{值}\text{近}\text{似}\text{代}\text{替}}\\{ \because \text{ }ds={ \left| {AB} \right| }=\sqrt{{ \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}} \left) \mathop{{}}\nolimits^{{2}}+ \left( y\mathop{{}}\nolimits_{{2}}-y\mathop{{}}\nolimits_{{1}} \left) \mathop{{}}\nolimits^{{2}}\right. \right. \right. \right. }}}\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }= \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}} \left) \sqrt{{1+ \left( \frac{{y\mathop{{}}\nolimits_{{2}}-y\mathop{{}}\nolimits_{{1}}}}{{x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}}}} \left) \mathop{{}}\nolimits^{{2}}\right. \right. }}\right. \right. }\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\sqrt{{1+ \left( \frac{{dy}}{{dx}} \left) \mathop{{}}\nolimits^{{2}}\right. \right. }}∙dx}\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\mathop{{ \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}} \right) }}\nolimits^{{\frac{{1}}{{2}}}}dx}\\\color{red}{ \therefore \text{ }\frac{{ds}}{{dx}}={ \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}}\right. }\mathop{ {\left) \right.} }\nolimits^{{\frac{{1}}{{2}}}}} \end{array}}$$

综上：

$${\begin{array}{*{20}{l}}{{{K=\mathop{{lim}}\limits_{{ \Delta s \to 0}}{ \left| \frac{{ \Delta \alpha }}{{ \Delta s}} \right| }=}{ \left| \frac{{d \alpha }}{{ds}} \right| }={ \left| {\frac{{d \alpha }}{{dx}}∙\frac{{1}}{{\frac{{ds}}{{dx}}}}} \right| }}}\\{\begin{array}{*{20}{l}}{\text{ }\text{ }\text{ }={ \left| {\frac{{{y '' }}}{{1+{y \prime }\mathop{{}}\nolimits^{{2}}}}∙\frac{{1}}{{{ \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}}\right. }\mathop{ {\left) \right.} }\nolimits^{{\frac{{1}}{{2}}}}}}} \right| }}\\\color{red}{{\text{ }\text{ }\text{ }=\frac{{{ \left| {{y '' }} \right| }}}{{{ \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}}\right. }\mathop{ {\left) \right.} }\nolimits^{{\frac{{3}}{{2}}}}}}}}\end{array}}\end{array}}$$

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若曲线y是由参数方程${ \left\{ {\begin{array}{*{20}{l}}{x= \varphi \left( t \right) }\\{y= \omega \left( t \right) }\end{array}}\right. }$确定的，则

$${\begin{array}{*{20}{l}}{{{y \prime }=\frac{{dy}}{{dx}}=\frac{{dy}}{{dt}}∙\frac{{1}}{{\frac{{dx}}{{dt}}}}}}\\{\text{ }\text{ }\text{ }\text{ }{=}\frac{{{ \omega \prime } \left( t \right) }}{{{ \varphi \prime } \left( t \right) }}}\\{{{{y '' }}=\frac{{d\mathop{{}}\nolimits^{{2}}y}}{{dx\mathop{{}}\nolimits^{{2}}}}=\frac{{d \left( \frac{{dy}}{{dx}} \right) }}{{dx}}=\frac{{d \left( \frac{{dy}}{{dx}} \right) }}{{dt}}∙\frac{{1}}{{\frac{{dx}}{{dt}}}}}}\\{\begin{array}{*{20}{l}}{\text{ }\text{ }\text{ }\text{ }\text{ }=\frac{{{{ \omega '' }} \left( t \left) { \varphi \prime } \left( t \left) -{ \omega \prime } \left( t \left) {{ \varphi '' }} \left( t \right) \right. \right. \right. \right. \right. \right. }}{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }}∙\frac{{1}}{{{ \varphi \prime } \left( t \right) }}}\\{\text{ }\text{ }\text{ }\text{ }\text{ }=\frac{{{{ \omega '' }} \left( t \left) { \varphi \prime } \left( t \left) -{ \omega \prime } \left( t \left) {{ \varphi '' }} \left( t \right) \right. \right. \right. \right. \right. \right. }}{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{3}}\right. \right. }}}\\{\text{曲}\text{率}K={\frac{{{ \left| {{y '' }} \right| }}}{{ \left( 1+\mathop{{{y \prime }}}\nolimits^{{2}} \left) \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}}}\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\frac{{{ \left| {\frac{{{{ \omega '' }} \left( t \left) { \varphi \prime } \left( t \left) -{ \omega \prime } \left( t \left) {{ \varphi '' }} \left( t \right) \right. \right. \right. \right. \right. \right. }}{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{3}}\right. \right. }}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }} \right| }}}{{ \left[ 1+\frac{{{ \omega \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }}{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }}\text{ }\text{ }\text{ } \left] \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}}\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\frac{{{ \left| {\frac{{{{ \omega '' }} \left( t \left) { \varphi \prime } \left( t \left) -{ \omega \prime } \left( t \left) {{ \varphi '' }} \left( t \right) \right. \right. \right. \right. \right. \right. }}{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{3}}\right. \right. }}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }} \right| }}}{{ \left[ \frac{{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }+{{ \omega \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }}}{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \left] \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}}\\{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\frac{{{ \left| {\frac{{{ \omega ''} \left( t \left) { \varphi \prime } \left( t \left) -{ \omega \prime } \left( t \left) { \varphi ''} \left( t \left) \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\right. \right. \right. \right. \right. \right. \right. \right. }}{{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{3}}\right. \right. }}∙{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{3}}\right. \right. }\text{ }\text{ }} \right| }}}{{ \left[ {{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }+{{ \omega \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }\text{ }\text{ }} \left] \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}}\\\color{red}{\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\frac{{ \left| {{{{ \omega '' }} \left( t \left) { \varphi \prime } \left( t \left) -{ \omega \prime } \left( t \left) {{ \varphi '' }} \left( t \left) \text{ }\right. \right. \right. \right. \right. \right. \right. \right. }} \right| }}{{ \left[ {{{ \varphi \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }+{{ \omega \prime } \left( t \left) \mathop{{}}\nolimits^{{2}}\right. \right. }\text{ }} \left] \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}}\end{array}}\end{array}}$$

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2.曲率半径的推导

  曲率半径用$\color{red}\rho$或者R表示。

$${{ \rho =\frac{{1}}{{K}}=}\frac{{ \left( 1+{y \prime }\mathop{{}}\nolimits^{{2}} \left) \mathop{{}}\nolimits^{{\frac{{3}}{{2}}}}\right. \right. }}{{ \left| {{y '' }} \right| }}}$$