常见的空间曲面及其对应的方程

曲面名称	图例	直角坐标系方程	参数方程
椭球面		${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$	$\{\begin{array}{l}x=a\sin u\cos v\\ y=\sin u\sin v\\ z=c\cos u\\ u\in [0,\pi ],v\in [0,2\pi ]\end{array}$
椭圆锥面		${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=z^2$	$\{\begin{array}{l}x=av\cos u\\ y=bv\sin u\\ z=cv\\ u\in [0,2\pi ],v\in \mathbb{R}\end{array}$
椭圆抛物面		${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=z$	$\{\begin{array}{l}x=av\cos u\\ y=bv\sin u\\ z=u^2\\ u\in [0,2\pi ],v\in [0,+\mathrm{\infty })\end{array}$
双曲抛物面		${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=z$	$\{\begin{array}{l}x=a(u+v)\\ y=b(u-v)\\ z=4uv\\ u,v\in \mathbb{R}\end{array}$
单叶双曲面		${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}-{\displaystyle \frac{z^2}{c^2}}=1$	$\{\begin{array}{l}x=a\cosh u\cos v\\ y=b\cosh u\sin v\\ z=c\sinh u\\ u\in \mathbb{R},v\in [0,2\pi ]\end{array}$
双叶双曲面		${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}-{\displaystyle \frac{z^2}{c^2}}=-1$	$\{\begin{array}{l}x=a\sqrt{u^2-1}\cos v\\ y=b\sqrt{u^2-1}\sin v\\ z=cu\\ u\in (-\mathrm{\infty },-1]\cup [1,+\mathrm{\infty }),v\in \mathbb{R}\end{array}$