二次曲线的切线（极线）方程

郝酒

已知二次曲线的方程为：$Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$，有一点为$(x_0,y_0)$，则$(x_0,y_0)$对应的极线方程为$
Ax_0x+B(x_0y+xy_0)+Cy_0y+D(x+x_0)+E(y+y_0)+F=0$，可以看成是如下矩阵的形式：
$$
(x,y,1)\left(\begin{array}\\A&B&D\\B&C&E\\D&E&F\end{array}\right)\left(\begin{array}\\x\\y\\1\end{array}\right)=0$$
极线为：
$$
(x,y,1)\left(\begin{array}\\A&B&D\\B&C&E\\D&E&F\end{array}\right)\left(\begin{array}\\x_0\\y_0\\1\end{array}\right)=0$$
相切的情形可以用隐函数求导验证，一般的情况，有没有一种看法，能得到上面的结论？感觉跟射影几何有关。

hbghlyj

Fishback, W.T. (1962), Projective and Euclidean Geometry (first edition), Wiley, pp. 164 9.5 Polarities

Definition 9.51. Two points are conjugate with respect to the conic with equation $∑ a_{i j} x_i x_j=0$ if their coordinates satisfy the condition $∑ a_{i j} x_i y_j=0$.

As an immediate consequence of this definition we have

Theorem 9.51. A point is self-conjugate with respect to a conic if and only if it lies on the conic.

Let us now fix a point $\left(x_1, x_2, x_3\right)$ and seek the locus of points conjugate to it. If the point were $(2,3,7)$ and the conic were
\[
x_1^2+x_1 x_3+x_3 x_1-2 x_2^2+4 x_3{ }^2=0,
\]
a point $\left(y_1, y_2, y_3\right)$ would have to satisfy
\[
2 y_1+2 y_3+7 y_1-6 y_2+28 y_3=0
\]
or
\[
9 y_1-6 y_2+30 y_3=0 \text {. }
\]
Thus the locus of points conjugate to $(2,3,7)$ would be a line. In general the locus of points conjugate to $\left(x_1, x_2, x_3\right)$ with respect to the conic with equation $∑ a_{i j} x_i x_j=0$ would be a line since the resulting bilinear form $∑ a_{i j} x_i y_j$ is linear in the $y$ 's. It is possible that all of the coefficients of the $y$ 's could be zero. In this case the locus of conjugate points is the entire plane. This would be the case for the point $(0,0,7)$ with respect to the conic
\[
x_1 x_2+x_2 x_1=0
\]
since the corresponding bilinear condition leads to
\[
0 y_1+0 y_2+0 y_3=0 .
\]
We summarize our results in the following theorem:

Theorem 9.52. The locus of all points conjugate to a given point with respect to a conic is a line or the entire plane.

In view of this result we make the following definition:

Definition 9.52. If the locus of all points conjugate to a fixed point with respect to a conic is a line, the line is the polar line of the fixed point with respect to the conic, and the fixed point is the pole of the line with respect to the conic. The relation between lines and points so determined is a polarity.

A polarity need not be defined everywhere since we have seen that the set of conjugate points of a given point may be the entire plane, in which case the point has no polar line. In the example above this occurred when the conic was degenerate. That this was no coincidence is shown in

Theorem 9.53. The polarity defined by a nondegenerate conic is a one-to-one correspondence between all of the points and all of the lines of the plane of the conic. ^[5]

Proof. Let the conic have equation $∑ a_{i j} x_i x_j=0$. Given a point $\left(x_1, x_2, x_3\right)$, the line coordinates $\left[u_1, u_2, u_3\right]$ of its polar line, if it exists, must be given by ^[6]
\begin{aligned}
&u_1=a_{11} x_1+a_{21} x_2+a_{31} x_3 \\
&u_2=a_{12} x_1+a_{22} x_2+a_{32} x_3 \\
&u_3=a_{13} x_1+a_{23} x_2+a_{33} x_3 .
\end{aligned}

---

^[5] The result is valid even if the conic is imaginary.
^[6] These equations establish a one-to-one correspondence between the points and the lines of a plane with coordinates linearly related. Thus a polarity is a projective transformation in the sense of Definition 8.72.

hbghlyj

Since the conic is nondegenerate, we know by Corollary 9.451 that
\[
\left|\begin{array}{lll}
a_{11} & a_{21} & a_{31} \\
a_{12} & a_{22} & a_{32} \\
a_{13} & a_{23} & a_{33}
\end{array}\right| \neq 0 .
\]
Now for a given point not all $x_i$ are zero, hence not all the $u_j$ can be zero, for if they were, equations (9.51) would have the unique solution $x_1=x_2=x_3=0$. We see each point has a unique polar line. Conversely, given any line, equations (9.51) become a system of equations for the coordinates of its pole. In view of the nonvanishing of the determinant of coefficients, this system will have a unique solution, and clearly not all of the $x_i$ will be zero.

Let us now investigate the relation between a point and its polar line. Let us suppose first that the point is on the conic, which is nondegenerate, and denote the coordinates of the point by the column matrix $X$.

Figure 9.17

The condition that a point with coordinates given by the column matrix $Y$ lie on the polar line is $X^* A Y=(0)$. We know by Theorem 9.51 that the given point is on its own polar line; that is, $X^* A X=(0)$. Suppose now we have a second point on the conic and on the polar line. If $Z$ is the column matrix of coordinates of this point, then $Z^* A Z=(0)$ and $X^* A Z=(0)$. Now any point on the polar line has a column matrix of coordinates of the form $c_1 X+c_2 Z$, and we know
\[
X^* A\left(c_1 X+c_2 Z\right)=c_1 X^* A X+c_2 X^* A Z=(0) .
\]
This polar line is also the polar line of the point with coordinates $Z$ since this line contains two points conjugate to the point with coordinates $Z$ (see Exercise 9.51). Thus $Z^* A\left(c_1 X+c_2 Z\right)=(0)$. It follows that
\[
\left(c_1 X+c_2 Z\right)^* A\left(c_1 X+c_2 Z\right)=(0) .
\]
This says every point on the polar line would be on the conic. This is impossible for a nondegenerate conic, and we conclude the polar line is the tangent line to the conic. We have proved

Theorem 9.54. The polar line to a point on a nondegenerate conic is the tangent line to the conic at that point.

We conclude by considering the relation for points not on a conic.

Theorem 9.55. If through a point not on a conic two distinct tangents to the conic exist, the polar line of the point is the line joining the points of tangency of the tangents.

Figure 9.18

Proof. In view of Theorems 9.16 and 9.39 there are no more than two tangents from such a point. The existence of tangents implies the conic is nondegenerate; hence every point has a polar line. Let the points of tangency be $P$ and $Q$. The given point is conjugate to each of these by Theorem 9.54. Conjugacy is symmetric (see Exercise 9.51); hence $P$ and $Q$ are conjugate to the given point and lie on its polar line.

Theorem 9.56. The polar line of a point with respect to a nondegenerate conic is the locus of points of intersection of tangents drawn from the intersections with the conic of lines through the given point.

Figure 9.19

Proof. In view of Theorem 9.55 any point on this locus is conjugate to the given point. Thus the line is the desired polar line.

hbghlyj

按照1#的写法

1. \begin{array}\\
2. 1\\
3. 2
4. \end{array}

复制代码

通过MathJax:\begin{array}\\
1\\
2
\end{array}
通过TeXLive:
! LaTeX Error: Illegal character in array arg.

kuing

hbghlyj 发表于 2022-12-12 17:27
按照1#的写法
通过MathJax:\begin{array}\\
1\\

缺了对齐方式的必要参数，加个 {c} ，比如

1. \begin{array}{c}\\
2. ...

复制代码

这样是没问题嘀。
Mathjax 一向“智能容错”。