# 漸伸線

${\displaystyle s\to r(s)+{r''(s) \over |r''(s)|^{2}}}$

## 參數化曲線

${\displaystyle X[x,y]=x-{\frac {x'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}}$

${\displaystyle Y[x,y]=y-{\frac {y'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}}$

## 範例

 圓的漸伸線(反向, by unwinding) 懸鏈線的漸開線是一個 曳物線。

### 圓的漸伸線

${\displaystyle \,x=a\left(\cos \ t+t\sin \ t\right)}$
${\displaystyle \,y=a\left(\sin \ t-t\cos \ t\right)}$

• 極坐標系中， ${\displaystyle \,r,\theta }$  一個圓的漸開線的參數方程可以寫成：
${\displaystyle \,r=a\sec \alpha }$
${\displaystyle \,\theta =\tan \alpha -\alpha }$

${\displaystyle \,r=a{\sqrt {1+t^{2}}}}$
${\displaystyle \,\theta =\arctan {\frac {\cos t+t\sin t}{\sin t-t\cos t}}}$ .

### 懸鏈線的漸開線

${\displaystyle x=t-\mathrm {tanh} (t)\,}$

${\displaystyle y=\mathrm {sech} (t)\,}$

${\displaystyle r(s)=(\sinh ^{-1}(s),\cosh(\sinh ^{-1}(s)))\,}$

${\displaystyle r(t)-tr^{\prime }(t)=(\sinh ^{-1}(t)-t/{\sqrt {1+t^{2}}},1/{\sqrt {1+t^{2}}})}$

### 擺線的漸開線

${\displaystyle x=r(t-\sin(t))\,}$
${\displaystyle y=r(1-\cos(t))\,}$