二次曲面

二次曲面的方程为：

${\displaystyle Q=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0\right\}}$

未退化的一般实二次曲面
橢球面	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1\,}$
橢圓拋物面	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-z=0\,}$
雙曲拋物面	${\displaystyle {x^{2} \over a^{2}}-{y^{2} \over b^{2}}-z=0\,}$
單葉雙曲面	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1\,}$
雙葉雙曲面	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1\,}$
退化的二次曲面
椭圆锥面	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0\,}$
橢圓柱面	${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}=1\,}$
雙曲柱面	${\displaystyle {x^{2} \over a^{2}}-{y^{2} \over b^{2}}=1\,}$
拋物柱面	${\displaystyle x^{2}+2ay=0\,}$

1. （页面存档备份，存于）, Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).

• 埃里克·韦斯坦因. . MathWorld.