查看“二次曲面”的源代码

[[File:Eccentricity.svg|链接=https://en.wikipedia.org/wiki/File:Eccentricity.svg|缩略图|285x285像素|有固定[[焦点 (几何)]] ''F'' 和準線的<span style="color:red;">橢圓形 (''e''&nbsp;=&nbsp;1/2)</span>，<span style="color:#00cc00;">拋物線(''e''&nbsp;=&nbsp;1)</span> 和 <span style="color:blue;">雙曲線(''e''&nbsp;=&nbsp;2)</span>。]]
'''二次曲面'''（Quadrics）指任何''n''維的超[[曲面]]，其定義為多元[[二次方程]]的解的軌跡。

在坐标<math>\{x_0, x_1, x_2, \ldots, x_D\}</math>，二次曲面的定義為代數方程<ref name="geom"> [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html], ''Quadrics'' in ''Geometry Formulas and Facts'' by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).</ref>
：

:<math>
\sum_{i,j=0}^D Q_{i,j}  x_i  x_j + \sum_{i=0}^D P_i  x_i + R = 0
</math>。

上式亦可以用[[矩陣乘法]]和[[向量]]的[[內積]]等概念，寫成以下形式：

:<math>\mathbf{x} =
\begin{pmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n \\
\end{pmatrix};
</math>　<math>
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};
</math>　<math>
\mathbf{b} =
\begin{pmatrix}
b_1 \\
b_2 \\
\vdots \\
b_n
\end{pmatrix}
</math>

:<math>\langle A\mathbf{x},\mathbf{x}\rangle+\langle\mathbf{b},\mathbf{x}\rangle+c=0</math>

二次曲面是[[代數簇]]的一種。

==欧几里得空间==
:{| class="wikitable" 
! colspan="3" style="background-color: white;" |未退化的一般实二次曲面
|-
|[[橢球]]面
|<math>{x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,</math>
|[[File:Ellipsoid_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Ellipsoid_Quadric.png|166x166像素]]
|-
|橢圓[[拋物面]]
|<math>{x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,</math>
|[[File:Paraboloid_Quadric.Png|链接=https://en.wikipedia.org/wiki/File:Paraboloid_Quadric.Png|166x166像素]]
|-
|雙曲[[拋物面]]
|<math>{x^2 \over a^2} - {y^2 \over b^2} - z = 0  \,</math>
|[[File:Hyperbolic_Paraboloid_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Hyperbolic_Paraboloid_Quadric.png|166x166像素]]
|-
|單葉[[雙曲面]]
|<math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,</math>
|[[File:Hyperboloid_Of_One_Sheet_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Hyperboloid_Of_One_Sheet_Quadric.png|163x163像素]]
|-
|双叶[[雙曲面|双曲面]]
|<math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,</math>
|[[File:Hyperboloid_Of_Two_Sheets_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Hyperboloid_Of_Two_Sheets_Quadric.png|163x163像素]]
|-
! colspan="3" style="background-color: white;" |特殊的二次曲面
|-
|[[類球面]]（一种特殊的[[橢球]]面）
|<math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,</math>
|[[File:Oblate_Spheroid_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Oblate_Spheroid_Quadric.png|83x83像素]][[File:Prolate_Spheroid_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Prolate_Spheroid_Quadric.png|83x83像素]]
|-
|[[球面]]（一种特殊的[[類球面]]）
|<math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,</math>
|[[File:Sphere_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Sphere_Quadric.png|166x166像素]]
|-
|圓[[拋物面]]（一种特殊的橢圓[[拋物面]]）
|<math>{x^2 \over a^2} + {y^2 \over a^2} - z = 0  \,</math>
|[[File:Circular_Paraboloid_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Circular_Paraboloid_Quadric.png|166x166像素]]
|-
|单叶[[雙曲面|双曲面]]（一种特殊的單葉[[雙曲面]]）
|<math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 1 \,</math>
|[[File:Circular_Hyperboloid_Of_One_Sheet_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Circular_Hyperboloid_Of_One_Sheet_Quadric.png|163x163像素]]
|-
|雙葉[[雙曲面]]（一種特殊的雙葉[[雙曲面]]）
|<math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = -1 \,</math>
|[[File:Circular_Hyperboloid_of_Two_Sheets_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Circular_Hyperboloid_of_Two_Sheets_Quadric.png|163x163像素]]
|-
! colspan="3" style="background-color: white;" |退化的二次曲面
|-
|椭圆锥面
|<math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,</math>
|[[File:Elliptical_Cone_Quadric.Png|链接=https://en.wikipedia.org/wiki/File:Elliptical_Cone_Quadric.Png|166x166像素]]
|-
|[[錐面]]
|<math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 0 \,</math>
|[[File:Circular_Cone_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Circular_Cone_Quadric.png|166x166像素]]
|-
|橢圓[[柱面]]
|<math>{x^2 \over a^2} + {y^2 \over b^2} = 1 \,</math>
|[[File:Elliptic_Cylinder_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Elliptic_Cylinder_Quadric.png|166x166像素]]
|-
|圓[[柱面]]（一种特殊的橢圓[[柱面]]）
|<math>{x^2 \over a^2} + {y^2 \over a^2} = 1  \,</math>
|[[File:Circular_Cylinder_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Circular_Cylinder_Quadric.png|166x166像素]]
|-
|雙曲[[柱面]]
|<math>{x^2 \over a^2} - {y^2 \over b^2} = 1 \,</math>
|[[File:Hyperbolic_Cylinder_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Hyperbolic_Cylinder_Quadric.png|166x166像素]]
|-
|拋物[[柱面]]
|<math>x^2 + 2ay = 0 \,</math>
|[[File:Parabolic_Cylinder_Quadric.png|链接=https://en.wikipedia.org/wiki/File:Parabolic_Cylinder_Quadric.png|166x166像素]]
|}

==参考来源==

<references />

== 外部链接 ==
*{{mathworld|urlname=Quadric|title=Quadric}} 

[[Category:曲面]]
[[Category:二次曲面|*]]
[[Category:解析几何|*]]