# 二次曲面

$\sum_{i,j=0}^D Q_{i,j} x_i x_j + \sum_{i=0}^D P_i x_i + R = 0$

$\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \\ \end{pmatrix};$　$A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{12} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{nn} \\ \end{pmatrix};$　$\mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix}$
$\langle A\mathbf{x},\mathbf{x}\rangle+2\langle\mathbf{b},\mathbf{x}\rangle+c=0$

## 例子

 橢球面 ${x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,$ 類球面 ${x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,$ 球面 ${x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,$ 橢圓拋物面 ${x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,$ 圓拋物面 ${x^2 \over a^2} + {y^2 \over a^2} - z = 0 \,$ 雙曲拋物面 ${x^2 \over a^2} - {y^2 \over b^2} - z = 0 \,$ 單葉雙曲面 ${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,$ 雙葉雙曲面 ${x^2 \over a^2} - {y^2 \over b^2} - {z^2 \over c^2} = 1 \,$ 錐面 ${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,$ 橢圓柱面 ${x^2 \over a^2} + {y^2 \over b^2} = 1 \,$ 圓柱面 ${x^2 \over a^2} + {y^2 \over a^2} = 1 \,$ 雙曲柱面 ${x^2 \over a^2} - {y^2 \over b^2} = 1 \,$ 拋物柱面 $x^2 + 2ay = 0 \,$

## 引用

1. ^ [1], Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).