# 曲率半径

## 定义

${\displaystyle R=\left\vert {ds \over d\varphi }\right\vert ={1 \over \kappa }}$

κ曲率

## 公式

### 二维

${\displaystyle R=\left|{\frac {\left(1+y'^{\,2}\right)^{\frac {3}{2}}}{y''}}\right|\,,}$

${\displaystyle R=\left|{\frac {ds}{d\varphi }}\right|=\left|{\frac {\left({{\dot {x}}^{2}+{\dot {y}}^{2}}\right)^{\frac {3}{2}}}{{\dot {x}}{\ddot {y}}-{\dot {y}}{\ddot {x}}}}\right|}$

${\displaystyle R={\frac {\left|\mathbf {v} \right|^{3}}{\left|\mathbf {v} \times \mathbf {\dot {v}} \right|}}\,,}$

${\displaystyle \left|\mathbf {v} \right|={\big |}({\dot {x}},{\dot {y}}){\big |}=R{\frac {d\varphi }{dt}}\,.}$

### n维

γ : ℝ → ℝnn中的参数方程曲线，则曲线上每个点的曲率半径ρ : ℝ → ℝ ，由[3]此可知

${\displaystyle \rho ={\frac {\left|{\boldsymbol {\gamma }}'\right|^{3}}{\sqrt {\left|{\boldsymbol {\gamma }}'\right|^{2}\,\left|{\boldsymbol {\gamma }}''\right|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}\,.}$

${\displaystyle \rho (t)={\frac {\left|1+f'^{\,2}(t)\right|^{\frac {3}{2}}}{\left|f''(t)\right|}}.}$

### 推导过程

γ如上，并固定t 。我们想要找到一个与t处的γ零阶、一阶和二阶导数相匹配的参数方程圆的半径ρ 。显然，半径与位置γ(t) 无关，而与速度γ′(t)和加速度γ″(t) 有关。 由向量vw只能获得三个独立标量，即v · vv · ww · w 。因此，曲率半径一定是关于这三个标量函数。即 |γ′(t)|2, |γ″(t)|2γ′(t) · γ″(t)[3]

n中圆的一般参数方程为

${\displaystyle \mathbf {g} (u)=\mathbf {a} \cos(h(u))+\mathbf {b} \sin(h(u))+\mathbf {c} }$

g的相关导数为

{\displaystyle {\begin{aligned}|\mathbf {g} '|^{2}&=\rho ^{2}(h')^{2}\\\mathbf {g} '\cdot \mathbf {g} ''&=\rho ^{2}h'h''\\|\mathbf {g} ''|^{2}&=\rho ^{2}\left((h')^{4}+(h'')^{2}\right)\end{aligned}}}

{\displaystyle {\begin{aligned}|{\boldsymbol {\gamma }}'(t)|^{2}&=\rho ^{2}h'^{\,2}(t)\\{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t)&=\rho ^{2}h'(t)h''(t)\\|{\boldsymbol {\gamma }}''(t)|^{2}&=\rho ^{2}\left(h'^{\,4}(t)+h''^{\,2}(t)\right)\end{aligned}}}

${\displaystyle \rho (t)={\frac {\left|{\boldsymbol {\gamma }}'(t)\right|^{3}}{\sqrt {\left|{\boldsymbol {\gamma }}'(t)\right|^{2}\,\left|{\boldsymbol {\gamma }}''(t)\right|^{2}-{\big (}{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t){\big )}^{2}}}}\,,}$

${\displaystyle \rho ={\frac {\left|{\boldsymbol {\gamma }}'\right|^{3}}{\sqrt {\left|{\boldsymbol {\gamma }}'\right|^{2}\;\left|{\boldsymbol {\gamma }}''\right|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}\,.}$

## 示例

### 半圆与圆

{\displaystyle {\begin{aligned}y&={\sqrt {a^{2}-x^{2}}}\\y'&={\frac {-x}{\sqrt {a^{2}-x^{2}}}}\\y''&={\frac {-a^{2}}{\left(a^{2}-x^{2}\right)^{\frac {3}{2}}}}\,.\end{aligned}}}

### 椭圆

${\displaystyle R(t)={\frac {(b^{2}\cos ^{2}t+a^{2}\sin ^{2}t)^{3/2}}{ab}}\,,}$

${\displaystyle R(\theta )={\frac {a^{2}}{b}}{\biggl (}{\frac {1-e^{2}(2-e^{2})(\cos \theta )^{2})}{1-e^{2}(\cos \theta )^{2}}}{\biggr )}^{3/2}\,,}$

${\displaystyle e^{2}=1-{\frac {b^{2}}{a^{2}}}\,.}$

## 參考

1. ^ Weisstien, Eric. Radius of Curvature. Wolfram Mathworld. [15 August 2016].
2. Kishan, Hari. Differential Calculus. Atlantic Publishers & Dist. 2007. ISBN 9788126908202 （英语）.
3. Love, Clyde E.; Rainville, Earl D. Differential and Integral Calculus Sixth. New York: MacMillan. 1962 （英语）.
4. ^ Weisstein, Eric W. Ellipse. mathworld.wolfram.com. [2022-02-23] （英语）.