抛物线的经过一点的法线个数

hbghlyj

Parabola Evolute
P为平面上一点, 证明抛物线$x=y^2$的经过一点P的法线个数是1,2或3, 分别当P在渐屈线左,上或右.

在Asymptote取等距的参数time为{0,10,20,30,40,50,60,70,80,90,100}作法线, 看上去有一些三线共点? 是精确的吗

hbghlyj

The equation of the normal at $(t^2,t)$ is
$$y-t = -2t(x-t^2)$$
Suppose $P(x_0,y_0)$ is on the normal
\begin{equation}\label1y_0-t = -2t(x_0-t^2)\end{equation}
This is a cubic equation in $t$. Computing discriminant:
$$\frac\Delta8=(2 x_0 - 1)^3 - \frac{27}2 y_0^2$$So $\Delta=0$ is the equation of the evolute.
If $\Delta>0$, \eqref{1} has 3 distinct real roots, P is on the left of the evolute;
If $\Delta=0$, \eqref{1} has 3 real roots and 2 of them are equal, P is on the evolute (the repeated root corresponds to the normal at P, the other root corresponds to the normal at another point);
If $\Delta<0$, \eqref{1} has 1 real root, P is on the right of the evolute.

青青子衿

hbghlyj 发表于 2023-2-24 20:33
The equation of the normal at $(t^2,t)$ is
$$y-t = -2t(x-t^2)$$
Suppose $P(x_0,y_0)$ is on the normal
\begin{align*}
y_0-t = -2t(x_0-t^2)
\end{align*}
This is a cubic equation in $t$. Computing discriminant:

确实和点到抛物线极小距离满足的三次方程有关
geogebra.org/m/JA9vHxWT

青青子衿

空间点到抛物面$z=Ax^2+By^2$的情况呢？

hbghlyj

hbghlyj 发表于 2023-2-23 18:59
在Asymptote取等距的参数time为{0,10,20,30,40,50,60,70,80,90,100}作法线, 看上去有一些三线共点?

是精确的：因为\eqref{1}的二次项系数$=0$，所以当$t_1+t_2+t_3=0$，在$(t_i^2,t_i)$处的三条法线就会共点。

也可直接代入法线方程$y-t_i = -2t_i(x-t_i^2)$，即$2t_ix+y-2t_i^3-t_i = 0$
$$\begin{vmatrix}2t_1&1&-2t_1^3-t_1\\2t_2&1&-2t_2^3-t_2\\2t_3&1&-2t_3^3-t_3\end{vmatrix}=4 (t_1 - t_2)  (t_2 - t_3) (t_3 - t_1)(t_1 + t_2 + t_3)$$当$t_1+t_2+t_3=0$，在$(t_i^2,t_i)$处的三条法线就会共点。