二次曲面

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二次曲面指任何n維的超曲面,其定義為多元二次方程的解的軌跡。

在坐标\{x_0, x_1, x_2, \ldots, x_D\},二次曲面的定義為代數方程[1]


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

上式亦可以用矩陣乘法向量內積等概念,寫成以下形式:

\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

二次曲面是代數簇的一種。

例子[编辑]

橢球 {x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \, Quadric Ellipsoid.jpg
    類球面 {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \, OblateSpheroid.PNGProlateSpheroid.png
       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 \, Quadric Elliptic Paraboloid.jpg
    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  \, Quadric Hyperbolic Paraboloid.jpg
單葉雙曲面 {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \, Quadric Hyperboloid 1.jpg
雙葉雙曲面 {x^2 \over a^2} - {y^2 \over b^2} - {z^2 \over c^2} = 1 \, Quadric Hyperboloid 2.jpg
錐面 {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \, Quadric Cone.jpg
橢圓柱面 {x^2 \over a^2} + {y^2 \over b^2} = 1 \, Quadric Elliptic Cylinder.jpg
    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 \, Quadric Hyperbolic Cylinder.jpg
拋物柱面 x^2 + 2ay = 0 \, Quadric Parabolic Cylinder.jpg

引用[编辑]

  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).