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# 曲率

## 我们要做什么

• 曲线上某个点的曲率半径，简单地说，就是在那一点处最贴合该曲线的圆的半径。
• 曲率，用\kappa来表示，是一除以曲率半径。
• 在公式中，曲率被定义为单位切线矢量函数相对于弧长的导数的大小：
\kappa, equals, open vertical bar, open vertical bar, start fraction, d, T, divided by, d, s, end fraction, close vertical bar, close vertical bar

• 直觉告诉我们，单位切线向量会告诉你移动的方向，以及相对于每一小步d, s在曲线上移动的速度，这是一个很好的指标能表示你转动的快慢。

## 沿着曲线驾驶

\kappa, equals, start fraction, 1, divided by, R, end fraction

## 计算曲率

$\vec{\textbf{s}}(t) = \left[ \begin{array}{c} t - \sin(t) \\ 1 - \cos(t) \end{array} \right]$

\begin{aligned} \quad x(t) &= t - \sin(t) \\ y(t) &= 1 - \cos(t) \end{aligned}

#### 第2步：求$\dfrac{dT}{ds}$start fraction, d, T, divided by, d, s, end fraction

\kappa, equals, open vertical bar, open vertical bar, start fraction, d, T, divided by, d, s, end fraction, close vertical bar, close vertical bar

open vertical bar, open vertical bar, start fraction, d, T, divided by, d, s, end fraction, close vertical bar, close vertical bar, equals, start fraction, open vertical bar, open vertical bar, start fraction, d, T, divided by, d, t, end fraction, close vertical bar, close vertical bar, divided by, open vertical bar, open vertical bar, start fraction, d, start bold text, s, end bold text, with, vector, on top, divided by, d, t, end fraction, close vertical bar, close vertical bar, end fraction

## 求单位切线向量

$\vec{\textbf{s}}(t) = \left[ \begin{array}{c} t - \sin(t) \\ 1 - \cos(t) \end{array} \right]$

$\dfrac{d\vec{\textbf{s}}}{dt} = \left[ \begin{array}{c} \dfrac{d}{dt}(t - \sin(t)) \\ \\ \dfrac{d}{dt}(1 - \cos(t)) \end{array} \right] = \left[ \begin{array}{c} 1 - \cos(t) \\ \sin(t) \end{array} \right]$

\begin{aligned} \quad \left[ \begin{array}{c} 1 - \cos(\pi) \\ \sin(\pi) \end{array} \right] = \left[ \begin{array}{c} 2 \\ 0 \end{array} \right] \end{aligned}

\begin{aligned} \quad \vec{\textbf{s}}'(t) = \left[ \begin{array}{c} 1 - \cos(t) \\ \sin(t) \end{array} \right] \end{aligned}

$\vec{\textbf{v}} = \left[\begin{array} \\ 2 \\ 1 \end{array} \right]$

## 用单位切线向量求曲率

\begin{aligned} \quad T(t) = \dfrac{\vec{\textbf{s}}'(t)}{||\vec{\textbf{s}}'(t)||} \end{aligned}

\begin{aligned} \quad \kappa = \left|\left| \dfrac{dT}{ds} \right|\right| \end{aligned}

\kappa, equals, open vertical bar, open vertical bar, start fraction, d, T, divided by, d, s, end fraction, close vertical bar, close vertical bar, equals, start fraction, open vertical bar, open vertical bar, start fraction, d, T, divided by, d, t, end fraction, close vertical bar, close vertical bar, divided by, open vertical bar, open vertical bar, start fraction, d, start bold text, s, end bold text, with, vector, on top, divided by, d, t, end fraction, close vertical bar, close vertical bar, end fraction

## 示例：螺旋的曲率

\begin{aligned} \quad \vec{\textbf{v}}(t) = \left[ \begin{array}{c} \cos(t) \\ \sin(t) \\ t/5 \end{array} \right] \end{aligned}

#### 第1步：计算导数

\begin{aligned} \quad \vec{\textbf{v}}(t) = \left[ \begin{array}{c} \cos(t) \\ \sin(t) \\ t/5 \end{array} \right] \end{aligned}

## 总结

• 曲线上某个点的曲率半径，简单地说，就是在那一点处最贴合该曲线的圆的半径。
• 曲率，用\kappa来表示，是一除以曲率半径。
• 已知定义曲线的参数函数start bold text, s, end bold text, with, vector, on top，要求曲率：
• start bold text, s, end bold text, with, vector, on top的导数归一化，以求得单位切线向量：
\begin{aligned} \quad T(t) = \dfrac{\vec{\textbf{s}}'(t)}{||\vec{\textbf{s}}'(t)||} \end{aligned}
• 曲率被定义为该值关于弧长s的导数的大小。你可以照下面的方式来计算：
\kappa, equals, open vertical bar, open vertical bar, start fraction, d, T, divided by, d, s, end fraction, close vertical bar, close vertical bar, equals, start fraction, open vertical bar, open vertical bar, start fraction, d, T, divided by, d, t, end fraction, close vertical bar, close vertical bar, divided by, open vertical bar, open vertical bar, start fraction, d, start bold text, s, end bold text, with, vector, on top, divided by, d, t, end fraction, close vertical bar, close vertical bar, end fraction
• 直觉告诉我们，单位切线向量会告诉你移动的方向，以及相对于每一小步d, s在曲线上移动的速度，这是一个很好的指标能表示你转动的快慢。