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[解析几何] 二次曲面和二次曲线

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发表于 2017-11-9 20:53:10 | 显示全部楼层 |阅读模式
  记空间中二次曲面的一般方程为
$$F(x,y,z)=a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{13}xz+2a_{23}yz$$
$$+2a_{14}x+2a_{24}y+2a_{34}z+a_{44}=0,$$
  其中,$a_{11}$,$a_{22}$,$a_{33}$,$a_{12}$,$a_{13}$,$a_{23}$不全为$0$。
  记$F(x,y,z)$的二次部分为
$$\Phi(x,y,z)=a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{13}xz+2a_{23}yz。$$
  利用矩阵的乘法可以把$F(x,y,z)$,$\Phi(x,y,z)$写成下列形式
$$F(x,y,z)=(x,y,z,1)\left( {\begin{array}{*{20}{c}}
a_{11}&a_{12}&a_{13}&a_{14}\\
a_{21}&a_{22}&a_{23}&a_{24}\\
a_{31}&a_{32}&a_{33}&a_{34}\\
a_{41}&a_{42}&a_{43}&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
x\\
y\\
z\\
1
\end{array}} \right),$$
$$\Phi(x,y,z)=(x,y,z)\left( {\begin{array}{*{20}{c}}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right)。$$
  记
$$A=\left( {\begin{array}{*{20}{c}}
a_{11}&a_{12}&a_{13}&a_{14}\\
a_{21}&a_{22}&a_{23}&a_{24}\\
a_{31}&a_{32}&a_{33}&a_{34}\\
a_{41}&a_{42}&a_{43}&a_{44}
\end{array}} \right),$$
$$\overline A=\left( {\begin{array}{*{20}{c}}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}
\end{array}} \right),$$
  $A$,$\overline A$分别称为二次曲面$F(x,y,z)=0$和$\Phi(x,y,z)$的系数矩阵,它们是实对称的。
  记$\delta^T=(a_{14},a_{24},a_{34})$,$\alpha^T=(x,y,z)$,则$A$可以分块写成
$$A=\left( {\begin{array}{*{20}{c}}
\overline A&\delta\\
\delta^T&a_{44}
\end{array}} \right)。$$
  二次曲面的方程
$$F(x,y,z)=a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{13}xz+2a_{23}yz$$
$$+2a_{14}x+2a_{24}y+2a_{34}z+a_{44}=0$$
  可表示成
$$F(x,y,z)=(\alpha^T,1)\left( {\begin{array}{*{20}{c}}
\overline A&\delta\\
\delta^T&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha\\
1
\end{array}} \right)=0。$$
$\Phi(x,y,z)$可以表示成
$$\Phi(x,y,z)=\alpha^TA\alpha。$$
  记
$$\Phi_1(x,y,z)=a_{11}x+a_{12}y+a_{13}z,$$
$$\Phi_2(x,y,z)=a_{12}x+a_{22}y+a_{23}z,$$
$$\Phi_3(x,y,z)=a_{13}x+a_{23}y+a_{33}z,$$
$$\Phi_4(x,y,z)=a_{14}x+a_{24}y+a_{34}z,$$
  则有
$$\Phi(x,y,z)=x\Phi_1(x,y,z)+y\Phi_2(x,y,z)+z\Phi_3(x,y,z)。$$
  由代数知识知道,实对称矩阵可用正交矩阵对角化。即对实对称矩阵$\overline A$,存在正交矩阵$T$,使$T^T \overline A T$为对角矩阵,且对角线上的元素为$\overline A$的特征值$\lambda_1$,$\lambda_2$,$\lambda_3$,即方程
$$\det (\overline A-\lambda E)=0$$
的根,它们全为实数。因而有
$$T^T \overline A T=\left( {\begin{array}{*{20}{c}}
\lambda_1&&\\
&\lambda_2&\\
&&\lambda_3
\end{array}} \right)=\wedge。$$
  对二次曲面的方程
$$F(x,y,z)=(\alpha^T,1)\left( {\begin{array}{*{20}{c}}
\overline A&\delta\\
\delta^T&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha\\
1
\end{array}} \right)=0,$$
  我们作如下的右手直角坐标变换,保持原点不动,从旧坐标系$\sigma_1=\left\{O;e_1,e_2,e_3 \right\}$到新坐标系$\sigma_2=\left\{O;e‘_1,e’_2,e‘_3 \right\}$的过渡矩阵为$T$,即
$$\alpha=T\alpha',$$
$$\left( {\begin{array}{*{20}{c}}
\alpha\\
1
\end{array}} \right)=\left( {\begin{array}{*{20}{c}}
T&0\\
0&1
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha'\\
1
\end{array}} \right)。$$
  将
$$\left( {\begin{array}{*{20}{c}}
\alpha\\
1
\end{array}} \right)=\left( {\begin{array}{*{20}{c}}
T&0\\
0&1
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha'\\
1
\end{array}} \right)$$
  代入二次曲面的方程
$$F(x,y,z)=(\alpha^T,1)\left( {\begin{array}{*{20}{c}}
\overline A&\delta\\
\delta^T&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha\\
1
\end{array}} \right)=0$$
  中得
$$(\alpha^T,1)\left( {\begin{array}{*{20}{c}}
\overline A&\delta\\
\delta^T&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha\\
1
\end{array}} \right)$$
$$=(\alpha‘^T,1)\left( {\begin{array}{*{20}{c}}
T^T&0\\
0&1
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\overline A&\delta\\
\delta^T&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
T&0\\
0&1
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha’\\
1
\end{array}} \right)$$
$$=(\alpha'^T,1)\left( {\begin{array}{*{20}{c}}
T^T \overline A T&T^T \delta\\
\delta^T T&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha'\\
1
\end{array}} \right)$$
$$=(\alpha'^T,1)\left( {\begin{array}{*{20}{c}}
\wedge&T^T \delta\\
\delta^T T&a_{44}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
\alpha'\\
1
\end{array}} \right)$$
  记$\delta^T T=(a'_{14},a'_{24},a'_{34})$。因此经过直角坐标变换
$$\alpha=T\alpha',$$
  曲面方程变为
$$F'(x',y',z')=\lambda_1x'^2+\lambda_2y'^2+\lambda_3z'^2+2a'_{14}x'+2a'_{24}y'+2a'_{34}z'+a_{44}=0。$$
  由以上知道,我们总能找到适当的右手直角坐标系使二次曲面的方程具有
$$F'(x',y',z')=\lambda_1x'^2+\lambda_2y'^2+\lambda_3z'^2+2a'_{14}x'+2a'_{24}y'+2a'_{34}z'+a_{44}=0$$
  的形式。因而为简洁起见不妨设二次曲面的方程就是
$$F'(x',y',z')=\lambda_1x'^2+\lambda_2y'^2+\lambda_3z'^2+2a'_{14}x'+2a'_{24}y'+2a'_{34}z'+a_{44}=0$$
  的形式,并将方程中的符号“$'$”去掉。在
$$F'(x',y',z')=\lambda_1x'^2+\lambda_2y'^2+\lambda_3z'^2+2a'_{14}x'+2a'_{24}y'+2a'_{34}z'+a_{44}=0$$
  的基础之上,通过配方,再作移轴,就可将方程
$$F'(x',y',z')=\lambda_1x'^2+\lambda_2y'^2+\lambda_3z'^2+2a'_{14}x'+2a'_{24}y'+2a'_{34}z'+a_{44}=0$$
  进一步化简,并了解其所对应的曲面。

情形1 $\lambda_1$,$\lambda_2$,$\lambda_3$都不为$0$。作移轴
$$\left\{ \begin{array}{l}
x'=x+\frac{a_{14}}{\lambda_1},\\
y'=y+\frac{a_{24}}{\lambda_2},\\
z'=z+\frac{a_{34}}{\lambda_3},
\end{array} \right.$$
  则有
$$F(x,y,z)=\lambda_1(x+\frac{a_{14}}{\lambda_1})^2+\lambda_2(y+\frac{a_{24}}{\lambda_2})^2+\lambda_3(z+\frac{a_{34}}{\lambda_3})^2-\frac{a_{14}^2}{\lambda_1}-\frac{a_{24}^2}{\lambda_2}-\frac{a_{34}^2}{\lambda_3}+a_{44}=0。$$
  令常数项为$a'_{44}$,得
$$\lambda_1x'^2+\lambda_2y'^2+\lambda_3z'^2+a'_{44}=0。$$
(1)$\lambda_1\lambda_2\lambda_3a'_{44}>0$
$1^{\circ}$$\lambda_1$,$\lambda_2$,$\lambda_3$同号,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+1=0$。虚椭球面
$2^{\circ}$$\lambda_1$,$\lambda_2$,$\lambda_3$异号,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}-1=0$。单叶双曲面
(2)$\lambda_1\lambda_2\lambda_3a'_{44}<0$
$3^{\circ}$$\lambda_1$,$\lambda_2$,$\lambda_3$同号,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}-1=0$。椭球面
$4^{\circ}$$\lambda_1$,$\lambda_2$,$\lambda_3$异号,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}+1=0$。双叶双曲面
(3)$a'_{44}=0$
$5^{\circ}$$\lambda_1$,$\lambda_2$,$\lambda_3$同号,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=0$。一点
$6^{\circ}$$\lambda_1$,$\lambda_2$,$\lambda_3$异号,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$。二次锥面

情形2 $\lambda_1$,$\lambda_2$,$\lambda_3$中只有一个为$0$。不妨设$\lambda_3=0$,作移轴
$$\left\{ \begin{array}{l}
x'=x+\frac{a_{14}}{\lambda_1},\\
y'=y+\frac{a_{24}}{\lambda_2},\\
z'=z,\end{array} \right.$$
  则有
$$\lambda_1x'^2+\lambda_2y'^2+2a_{34}z'+a'_{44}=0。$$
(1)$a_{34} \ne 0$,再作移轴
$$\left\{ \begin{array}{l}
x'‘=x’,\\
y''=y',\\
z''=z'+\frac{a'_{44}}{2a_{34}},
\end{array} \right.$$
  那么
$$\lambda_1x'^2+\lambda_2y'^2+2a_{34}z'+a'_{44}=0$$
  化简为
$$\lambda_1x''^2+\lambda_2y''^2+2a_{34}z''=0$$
$7^{\circ}$$\lambda_1\lambda_2>0$,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}=2z$。椭圆抛物面
$8^{\circ}$$\lambda_1\lambda_2<0$,则同于形式$\frac{x^2}{a^2}-\frac{y^2}{b^2}=2z$。双曲抛物面
(2)$a_{34}=0$,$a'_{44} \ne 0$,则
$$\lambda_1x'^2+\lambda_2y'^2+2a_{34}z'+a'_{44}=0$$
  变为
$$\lambda_1x'^2+\lambda_2y'^2+a'_{44}=0$$
$9^{\circ}$$\lambda_1$,$\lambda_2$同号但与$a'_{44}$异号,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0$。椭圆柱面
$10^{\circ}$$\lambda_1$,$\lambda_2$,$a'_{44}$同号,则同于形式$\frac{x^2}{a^2}-\frac{y^2}{b^2}+1=0$。虚椭圆柱面
$11^{\circ}$$\lambda_1\lambda_2<0$,则同于形式$\frac{x^2}{a^2}-\frac{y^2}{b^2}-1=0$。双曲柱面
(3)$a_{34}=a'_{44}=0$
$12^{\circ}$$\lambda_1\lambda_2>0$,则同于形式$\frac{x^2}{a^2}+\frac{y^2}{b^2}=0$。一对相交于一条实直线的虚平面
$13^{\circ}$$\lambda_1\lambda_2<0$,则同于形式$\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$。一对相交平面

情形3 $\lambda_1$,$\lambda_2$,$\lambda_3$中有两个为$0$。不妨设$\lambda_1 \ne 0$,作移轴
$$\left\{ \begin{array}{l}
x'=x+\frac{a_{14}}{\lambda_1},\\
y'=y,\\
z'=z,\end{array} \right.$$
  则有
$$\lambda_1x'^2+2a_{24}y'+2a_{34}z'+a'_{44}=0。$$
(1)$a_{24}$,$a_{34}$中至少有一个不为$0$,作变换
$\left\{ \begin{array}{l}
x''=x',\\
y''=\frac{2a_{24}y'+2a_{34}z'+a'_{44}}{2\sqrt {a_{24}^2+a_{34}^2}},\\
z''=\frac{-a_{34}y'+a_{24}z'}{\sqrt {a_{24}^2+a_{34}^2}},\end{array} \right.$,
  通过此变换,
$$\lambda_1x'^2+2a_{24}y'+2a_{34}z'+a'_{44}=0$$
  可化简成如下形式:
$14^{\circ}$$x^2=2py$。抛物柱面
(2)$a_{24}=a_{34}=0$
$15^{\circ}$$\lambda_1$与$a'_{44}$异号,则同于形式$x^2-a^2=0$。一对平行平面
$16^{\circ}$$\lambda_1$与$a'_{44}$同号,则同于形式$x^2+a^2=0$。一对虚的平行平面
$17^{\circ}$$a'_{44}=0$,则同于形式$x^2=0$。一对重合平面
  综合以上结论,我们有下列定理。

定理1 选取适当的坐标系,二次曲面方程总可以化简为以下五个简化方程中的一个:
(1)$a_{11}x^2+a_{22}y^2+a_{33}z^2+a_{44}=0$,$a_{11}a_{22}a_{33} \ne 0$;
(2)$a_{11}x^2+a_{22}y^2+2a_{34}z=0$,$a_{11}a_{22}a_{34} \ne 0$;
(3)$a_{11}x^2+a_{22}y^2+a_{44}=0$,$a_{11}a_{22} \ne 0$;
(4)$a_{11}x^2+2a_{24}y=0$,$a_{11}a_{24} \ne 0$;
(5)$a_{11}x^2+a_{44}=0$,$a_{11} \ne 0$。
  二次曲面总共有$17$种曲面。

  类似于空间二次曲面的讨论,平面上的二次曲线方程有如下结论。
  记平面上的二次曲线方程为
$$F(x,y)=a_{11}x^2+a_{22}y^2+2a_{12}xy+2a_{13}x+2a_{23}y+a_{33}$$
$$=(x,y,1)\left( {\begin{array}{*{20}{c}}
a_{11}&a_{12}&a_{13}\\
a_{12}&a_{22}&a_{23}\\a_{13}&a_{23}&a_{33}\end{array}} \right)\left( {\begin{array}{*{20}{c}}
x\\
y\\1\end{array}} \right)=0。$$

  记
$$\Phi(x,y)=(x,y)\left( {\begin{array}{*{20}{c}}
a_{11}&a_{12}\\
a_{12}&a_{22}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
x\\
y
\end{array}} \right),$$
$$\Phi_1(x,y)=a_{11}x+a_{12}y,$$
$$\Phi_2(x,y)=a_{12}x+a_{22}y,$$
$$\Phi_3(x,y)=a_{13}x+a_{23}y。$$
  经过类似于二次曲面方程的化简过程可以得到以下定理。

定理1' 平面上的二次曲线方程可化简为以下三个简化方程之一:
(1)$a_{11}x^2+a_{22}y^2+a_{33}=0$,$a_{11}a_{22} \ne 0$;
(2)$a_{22}y^2+2a_{13}x=0$,$a_{22}a_{13} \ne 0$;
(3)$a_{22}y^2+a_{33}=0$,$a_{22} \ne 0$。

  二次曲线共分为$9$种,它们的方程形式如下:
$1^{\circ}$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0$,椭圆
$2^{\circ}$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+1=0$,虚椭圆
$3^{\circ}$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=0$,交于一实点的二条虚直线
$4^{\circ}$$\frac{x^2}{a^2}-\frac{y^2}{b^2}-1=0$,双曲线
$5^{\circ}$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$,两条相交直线
$6^{\circ}$$y^2-2px=0$,抛物线
$7^{\circ}$$y^2-a^2=0$,一对平行直线
$8^{\circ}$$y^2+a^2=0$,一对虚平行直线
$9^{\circ}$$y^2=0$。一对重合直线
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