# 參數方程

${\displaystyle {\begin{cases}x=f(t)\\y=g(t)\end{cases}}}$

## 例

${\displaystyle x=a\cos(t),y=a\sin(t)}$，表示了平面上半徑為${\displaystyle a}$、以原點為圓心的。在三維，加入${\displaystyle z=bt}$，便是螺旋的圖形。這些式子可以表示成：

${\displaystyle r(t)=(x(t),y(t),z(t))=(a\cos(t),a\sin(t),bt)\,}$

${\displaystyle v(t)=r'(t)=(x'(t),y'(t),z'(t))=(-a\sin(t),a\cos(t),b)\,}$

${\displaystyle a(t)=r''(t)=(x''(t),y''(t),z''(t))=(-a\cos(t),-a\sin(t),0)\,}$

${\displaystyle r(u,v)=(x(u,v),y(u,v),z(u,v))=(a\cos(u),a\sin(u),v)\,}$

## 常见参数方程

${\displaystyle {\begin{cases}x=r\cos t\\y=r\sin t\end{cases}}}$

• 直线
• 点斜式过${\displaystyle (x_{0},y_{0})}$斜率${\displaystyle m}$的直线: ${\displaystyle {\begin{cases}x=x_{0}+t\\y=y_{0}+mt\end{cases}}}$
• 点向式过${\displaystyle (x_{0},y_{0})}$, 方向向量${\displaystyle (u,v)}$的直线：${\displaystyle {\begin{cases}x=x_{0}+ut\\y=y_{0}+vt\end{cases}}}$
• ${\displaystyle {\begin{cases}x=r\cos t\\y=r\sin t\end{cases}}}$
• 椭圆${\displaystyle {\begin{cases}x=a\cos t\\y=b\sin t\end{cases}}}$
• 双曲线${\displaystyle {\begin{cases}x=a\sec t\\y=b\tan t\end{cases}}}$
• 抛物线${\displaystyle {\begin{cases}x=2ct\\y=t^{2}\end{cases}}}$
• 螺线${\displaystyle {\begin{cases}x=t\cos lt\\y=t\sin lt\end{cases}}}$
• 摆线${\displaystyle {\begin{cases}x=r\cdot \left(t-\sin t\right)\\y=r\cdot \left(1-\cos t\right)\end{cases}}}$