椭球面和三正交切线

数学星空

1.若已知原椭圆\(ax^2+by^2+cz^2=1\)上一点\(x_1,y_1,z_1\),试给出另外两点\(x_2,y_2,z_2,x_3,y_3,z_3\)及另一个顶点\(x_0,y_0,z_0\)所满足的代数方程？

2.若已知原椭圆\(ax^2+by^2+cz^2=1\)及轨迹曲面上的一点\(x_0,y_0,z_0\),试给出另外三点\(x_1,y_1,z_1,x_2,y_2,z_2,x_3,y_3,z_3\)所满足的代数方程？

hejoseph

数学星空 发表于 2018-11-30 18:53
在椭圆\(x^2+\frac{y^2}{2}+\frac{z^2}{4}=1\)曲面上中取了150个样本点\({x1,y1,z1}\)，最后只算出了其中70 ...

Maple我不是太懂，不知道哪些点算不出来。不过我想不是没有解，而是对于曲面 $ax^2+by^2+cz^2=1$ 上任意一点 $(x_1,y_1,z_1)$，对应轨迹曲面上的点 $(x_0,y_0,z_0)$ 的点一般是有无数个的，而且根据38#的讨论也知道对确定的 $(x_0,y_0,z_0)$，另外两点 $(x_2,y_2,z_2)$、$(x_3,y_3,z_3)$ 一定存在的。
例如对于 $a=1$，$b=1/2$，$c=1/4$，取 $(x_1,y_1,z_1)=(1/3,2/5,2\sqrt{182}/15)$，那么得到的 $(x_0,y_0,z_0)$ 中的 $x_0$ 满足 $-0.0783644\leq x_0\leq 1.26133$，任意取这个范围内的 $x_0$，都可以计算出 $y_0$，$z_0$，和另外两切点 $(x_2,y_2,z_2)$、$(x_3,y_3,z_3)$，我也在Mathematica中计算过了，确实是没问题的。

数学星空

你可以试一下下面30组，看能算出多少组实数解?
  \(a=1,b=\frac{1}{2},c=\frac{1}{4},x1=\frac{\sqrt{5}}{3}\frac{k}{30},y1=\frac{\sqrt{5}}{3}\sqrt{2(1-(\frac{k}{30})^2)},z1=\frac{4}{3},k=1...30\)

hejoseph

数学星空 发表于 2018-12-1 14:58
你可以试一下下面30组，看你要算出几组实数解
  \(a=1,b=1/2,c=1/4,x1=\frac{\sqrt{5}}{3}\frac{k}{30},y1 ...

我还是用我之前给的那组数据吧，上面提到过一点 $(x_1,y_1,z_1)$ 对应的 $(x_0,y_0,z_0)$ 并不唯一，你可以试试计算下的。对于 $a=1$，$b=1/2$，$c=1/4$，取 $(x_1,y_1,z_1)=(1/3,2/5,2\sqrt{182}/15)$，我计算验证四组数据，都是符合条件的
(x0,y0,z0)=(0,0.8938618351205301,1.8262032652997853)，(x2,y2,z2)=(-0.4254859412404586,0.6201784653036762,1.583225883307254)，(x3,y3,z3)=(0.27282627875537524,1.118615039870601,1.0952917746575854)
(x0,y0,z0)=(0.5,1.2090696352176804,1.3153900671122774)，(x2,y2,z2)=(0.8392306677320858,0.6708967941353519,0.5315661093313266)，(x3,y3,z3)=(-0.1298929660253607,1.1855882841629264,1.0589013695428644)
(x0,y0,z0)=(1,0.9672557081741443,1.0523120536898218)，(x2,y2,z2)=(0.20376541074777976,1.3164394710417282,0.6065417279115298)，(x3,y3,z3)=(0.9906370630640637,-0.10518828197420715,0.22896219732928225)
(x0,y0,z0)=(1.2,0.6419287569230474,1.0487512185413908)，(x2,y2,z2)=(0.4206260259119077,1.2593008082132435,0.34730092739476065)，(x3,y3,z3)=(0.9077606649755143,-0.5460756642177671,0.32784910918169313)

hejoseph

hejoseph 发表于 2018-12-1 15:15
我还是用我之前给的那组数据吧，上面提到过一点 $(x_1,y_1,z_1)$ 对应的 $(x_0,y_0,z_0)$ 并不唯一，你可 ...

回复数学星空，我的计算方法是这样的，对于曲面 $ax^2+by^2+cz^2=1$ 上的一点 $(x_1,y_1,z_1)$，通过方程组
\[
\left\{
\begin{aligned}
&ax_1x_0+by_1y_0+cz_1z_0=1\\
&a(b+c)x_0^2+b(a+c)y_0^2+c(a+b)z_0^2=a+b+c
\end{aligned}
\right.
\]
消去 $z_0$，得到一个关于 $y_0$ 的二次方程，从判别式可以得到一个关于 $x_0$ 的不等式，消去 $y_0$，得到一个关于 $z_0$ 的二次方程，从判别式可以得到一个关于 $x_0$ 的不等式，两个关于 $x_0$ 的不等式的公共解集可以确定 $x_0$ 的范围。在这个范围内任意取一个 $x_0$ 的值，再代入上面的方程组，即可求得 $y_0$、$z_0$ 的实数解。取一组 $(x_0,y_0,z_0)$，解这个方程组，便可得 $(x_2,y_2,z_2)$、$(x_3,y_3,z_3)$
\[
\begin{cases}
(x_1-x_0)(x-x_0)+(y_1-y_0)(y-y_0)+(z_1-z_0)(z-z_0)=0\\
ax_0x+by_0y+cz_0z=1\\
ax^2+by^2+cz^2=1
\end{cases}
\]

hejoseph

对于 $a>0$、$b>0$、$c>0$，曲面 $ax^2+by^2+cz^2=1$ 上一点 $(x_1,y_1,z_1)$，对应轨迹方程上的点为 $(x_0,y_0,z_0)$，从方程组
\[
\left\{
\begin{aligned}
&ax_1^2+by_1^2+cz_1^2=1\\
&ax_0x_1+by_0y_1+cz_0z_1=1
\end{aligned}
\right.
\]
消去 $z_1$，得到一个关于 $y_1$ 的二次方程，其判别式为
\[
-4bcz_0^2\Delta
\]
消去 $y_1$，得到一个关于 $z_1$ 的二次方程，其判别式为
\[
-4bcy_0^2\Delta
\]
其中
\[
\Delta=\left(a^2x_0^2+aby_0^2+acz_0^2\right)x_1^2-2ax_0x_1-by_0^2-cz_0^2+1
\]
如果存在某点 $(x_1,y_1,z_1)$ 对应的 $(x_0,y_0,z_0)$ 不存在，那么必须对任意轨迹曲面上的点 $(x_0,y_0,z_0)$，$\Delta$ 值都为正数，就是说
\[
\left(a^2x_0^2+aby_0^2+acz_0^2\right)x_1^2-2ax_0x_1-by_0^2-cz_0^2+1>0
\]
上式看作 $x_1$ 的二次函数，其最小值为
\[
M=-\frac{\left(by_0^2+cz_0^2\right)\left(ax_0^2+by_0^2+cz_0^2-1\right)}{ax_0^2+by_0^2+cz_0^2}
\]
不妨设 $c$ 最小，此时从轨迹方程
\[
a(b+c)x_0^2+b(a+c)y_0^2+c(a+b)z_0^2=a+b+c
\]
求出 $z_0^2$ 的值再代入 $ax_0^2+by_0^2+cz_0^2-1$ 中，得
\[
ax_0^2+by_0^2+cz_0^2-1=\frac{a(a-c)x_0^2+b(b-c)y_0^2+c}{a+b}>0
\]
也就是说必定存在轨迹上的点 $(x_0,y_0,z_0)$，使得 $\Delta\leq 0$，也就是说解必定是存在的。

葡萄糖

hejoseph 发表于 2018-11-30 10:18
命题1：平面 $Ax+By+Cz=0$ 截取锥面 $ax^2+by^2+cz^2=0$ 的两条直线的夹角余弦值为
\[
\frac{\left|A^2(b+c)+B^2(a+c)+C^2(a+b)\right|}{\sqrt{A^4(b-c)^2+B^4(a-c)^2+C^4(a-b)^2+2A^2B^2(a-c)(b-c)+2A^2C^2(a-b)(c-b)+2B^2C^2(b-a)(c-a)}}
\]
这个容易证明，不写过程了。
...

平面 $Ax+By+Cz=0$ 截取二次锥面 $axy+byz+czx=0$ 的两条直线的夹角余弦值为
？？？？？

为了更直观的理解第#38楼的定理
补个楼，贴个图，举例说明：过三条坐标轴的二次锥面还有其他三条两两正交的母线（该图并没有证明有无数组，但直观看出不止一组）


上图中（橙色）的曲面为二次锥面\(\,\Gamma\colon\,4xy+5yz+2zx=0\,\)，显然该二次锥面过\(x\)轴、\(y\)轴、\(z\)轴（黑色）
上有一条母线（红色直线）\(l_1\colon\{5t,-t,4t\}\)，过原点与该条母线垂直的（蓝色）平面\(\,l_2Ol_3\colon\,5x-y+4z=0\,\)
该平面与二次锥面交于两条直线\(\,l_2\colon\{-\frac{43+\sqrt{249}}{40}t,-\frac{11+\sqrt{249}}{8}t,t\}\,\)和\(\,l_3\colon\{-\frac{43-\sqrt{249}}{40}t,-\frac{11-\sqrt{249}}{8}t,t\}\,\)

1. Show[
   
2.    ContourPlot3D[{4 x y + 5 y z + 2 z x == 0,
   
3.    5 x - y + 4 z == 0}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
   
4.    Mesh -> None, ContourStyle -> Opacity[0.8]],
   
5.    ParametricPlot3D[{{5 t, -t, 4 t},
   
6.    {(-43 - Sqrt[249])/40 t, (-11 - Sqrt[249])/8 t,  t},
   
7.    {(-43 + Sqrt[249])/40 t, (-11 + Sqrt[249])/8 t, t}},
   
8.    {t, -1, 1}, PlotStyle -> Red],
   
9.    ParametricPlot3D[{{u, 0, 0}, {0, u, 0}, {0, 0, u}},
   
10.    {u, -1, 1}, PlotStyle -> Black]]

复制代码

这个二次锥面上会有无数组三正交母线也太神奇了！
二次锥面（quadric conical surface）其实就是 “椭圆锥面”
在没有看到这证明之前，一直以为有有限组，因此之前错误地认为椭球面三正交切线的轨迹是三维区域；
现在终于能直观的理解了：椭球面三正交切线的轨迹是椭球面

现在还遗留两个问题：
①椭球面张角固定的外切圆锥顶点的轨迹是？？？
②椭球面外棱切正四面体顶点的轨迹是？？？

【注】：外棱切正四面体顶点，即三切线相交于一点，两两互成60度角。

mathe

应该这样推广。给定一个二次曲面$\Gamma$, 其上三根切线$l_1,l_2,l_3$相交于点$P$,对应切点分别为$A,B,C$，必须满足内积\( \langle\overrightarrow{PA} ,\overrightarrow{PB} \rangle = \langle\overrightarrow{PB} ,\overrightarrow{PC}\rangle = \langle\overrightarrow{PC} ,\overrightarrow{PA}\rangle =h\),其中$h$是事先给定的常数，那么$P$点的轨迹必然是一个二次曲面。

而且三根切线我们还可以拓展成n条切线，即：
给定一个二次曲面$\Gamma$, 其上$n$根切线$l_1,l_2,...,l_n$相交于点$P$,对应切点分别为$A_1,A_2,...,A_n$，必须满足内积\( \langle\overrightarrow{PA_1} ,\overrightarrow{PA_2} \rangle =  \langle\overrightarrow{PA_2} ,\overrightarrow{PA_3} \rangle =  \langle\overrightarrow{PA_3} ,\overrightarrow{PA_4} \rangle = \cdots = \langle\overrightarrow{PA_{n-1}} ,\overrightarrow{PA_n} \rangle = \langle\overrightarrow{PA_n} ,\overrightarrow{PA_1} \rangle =h\),其中$h$是事先给定的常数，那么$P$点的轨迹必然是一个二次曲面。(当然通常n比较大时,$h$必须是比较小的正数值才可能有解）

hejoseph

数学星空 发表于 2018-12-1 14:58
你可以试一下下面30组，看能算出多少组实数解?
  \(a=1,b=\frac{1}{2},c=\frac{1}{4},x1=\frac{\sqrt{5}}{ ...

Mathematica中输入
a = 1;
b = 1/2;
c = 1/4;
z1 = 4/3;
For[k = 1, k <= 30, k++,
x1 = Sqrt[5]/3 k/30;
y1 = Sqrt[5]/3 Sqrt[2 (1 - (k/30)^2)];
Print[k];
Print[N[x1]];
Print[N[y1]];
Print[
  NSolve[
   {a (b + c) x0^2 + b (a + c) y0^2 + c (a + b) z0^2 == a + b + c,
    a x1 x0 + b y1 y0 + c z1 z0 == 1,
    a x2 x0 + b y2 y0 + c z2 z0 == 1,
    a x3 x0 + b y3 y0 + c z3 z0 == 1,
    a x2^1 + b y2^2 + c z2^2 == 1,
    a x3^1 + b y3^2 + c z3^2 == 1,
    (x1 - x0) (x2 - x0) + (y1 - y0) (y2 - y0) + (z1 - z0) (z2 - z0) ==
      0,
    (x1 - x0) (x3 - x0) + (y1 - y0) (y3 - y0) + (z1 - z0) (z3 - z0) ==
      0,
    (x2 - x0) (x3 - x0) + (y2 - y0) (y3 - y0) + (z2 - z0) (z3 - z0) ==
      0},
   {x0, y0, z0, x2, y2, z2, x3, y3, z3}, Reals
   ]
  ]
]
运行结果
1

0.0248452

1.05351

{{x0->0.317898,y0->0.659803,z0->1.93364,x2->0.199072,y2->0.283042,z2->1.74456,x3->-5.00169,y3->0.941536,z3->4.71527},{x0->0.592822,y0->1.45657,z0->0.654049,x2->0.716809,y2->0.717894,z2->0.319409,x3->-0.283691,y3->1.6023,z3->0.00762818},{x0->0.592822,y0->1.45657,z0->0.654049,x2->-0.283691,y2->1.6023,z2->0.00762818,x3->0.716809,y3->0.717894,z3->0.319409},{x0->0.317898,y0->0.659803,z0->1.93364,x2->-5.00169,y2->0.941536,z2->4.71527,x3->0.199072,y3->0.283042,z3->1.74456}}

2

0.0496904

1.05175

{{x0->0.33025,y0->0.640493,z0->1.94031,x2->0.199793,y2->0.268249,z2->1.7484,x3->-4.57201,y3->1.048,z3->4.48235},{x0->0.618195,y0->1.45463,z0->0.61299,x2->0.750615,y2->0.677341,z2->0.282766,x3->-0.228481,y3->1.56746,z3->0.00788712},{x0->0.618195,y0->1.45463,z0->0.61299,x2->-0.228481,y2->1.56746,z2->0.0078871,x3->0.750615,y3->0.677341,z3->0.282766},{x0->0.33025,y0->0.640493,z0->1.94031,x2->-4.57201,y2->1.048,z2->4.48235,x3->0.199793,y3->0.268249,z3->1.7484}}

3

0.0745356

1.04881

{{x0->0.341303,y0->0.620692,z0->1.9472,x2->0.200262,y2->0.253351,z2->1.75231,x3->-4.16795,y3->1.12556,z3->4.25888},{x0->0.641117,y0->1.45305,z0->0.570684,x2->0.780433,y2->0.639967,z2->0.24321,x3->-0.186142,y3->1.54022,z3->0.00230859},{x0->0.641117,y0->1.45305,z0->0.570684,x2->-0.186142,y2->1.54022,z2->0.00230859,x3->0.780433,y3->0.639967,z3->0.24321},{x0->0.341303,y0->0.620692,z0->1.9472,x2->-4.16795,y2->1.12556,z2->4.25888,x3->0.200262,y3->0.253351,z3->1.75231}}

4

0.0993808

1.04468

{{x0->0.351111,y0->0.600565,z0->1.95422,x2->0.200494,y2->0.238401,z2->1.75623,x3->-3.79561,y3->1.1796,z3->4.04963},{x0->0.661504,y0->1.45182,z0->0.527741,x2->0.805878,y2->0.606614,z2->0.201318,x3->-0.153906,y3->1.51914,z3->-0.0072109},{x0->0.351111,y0->0.600565,z0->1.95422,x2->-3.79561,y2->1.1796,z2->4.04963,x3->0.200494,y3->0.238401,z3->1.75623},{x0->0.661504,y0->1.45182,z0->0.527741,x2->-0.153906,y2->1.51914,z2->-0.0072109,x3->0.805878,y3->0.606614,z3->0.201318}}

5

0.124226

1.03935

{{x0->0.359743,y0->0.58026,z0->1.96129,x2->0.200503,y2->0.223442,z2->1.76015,x3->-3.45695,y3->1.21505,z3->3.85683},{x0->0.679361,y0->1.45091,z0->0.484817,x2->0.826887,y2->0.577732,z2->0.157815,x3->-0.12956,y3->1.50298,z3->-0.0191581},{x0->0.359743,y0->0.58026,z0->1.96129,x2->-3.45695,y2->1.21505,z2->3.85683,x3->0.200503,y3->0.223442,z3->1.76015},{x0->0.679361,y0->1.45091,z0->0.484817,x2->-0.12956,y2->1.50298,z2->-0.0191581,x3->0.826887,y3->0.577732,z3->0.157815}}

6

0.149071

1.0328

{{x0->0.367275,y0->0.559905,z0->1.96835,x2->0.200302,y2->0.208508,z2->1.76404,x3->-3.15129,y3->1.23609,z3->3.68094},{x0->0.694771,y0->1.45025,z0->0.442568,x2->0.843634,y2->0.553438,z2->0.113472,x3->-0.111367,y3->1.49071,z3->-0.0323286},{x0->0.367275,y0->0.559905,z0->1.96835,x2->-3.15129,y2->1.23609,z2->3.68094,x3->0.200302,y3->0.208508,z3->1.76404},{x0->0.694771,y0->1.45025,z0->0.442568,x2->-0.111367,y2->1.49071,z2->-0.0323286,x3->0.843634,y3->0.553438,z3->0.113472}}

7

0.173916

1.025

{{x0->0.707885,y0->1.44979,z0->0.401611,x2->0.856444,y2->0.5336,z2->0.0690312,x3->-0.0979775,y3->1.48152,z3->-0.0457618},{x0->0.37379,y0->0.539608,z0->1.97533,x2->0.199904,y2->0.193621,z2->1.76788,x3->-2.87641,y3->1.24611,z3->3.52138},{x0->0.37379,y0->0.539608,z0->1.97533,x2->-2.87641,y2->1.24611,z2->3.52138,x3->0.199904,y3->0.193621,z3->1.76788},{x0->0.707885,y0->1.44979,z0->0.401611,x2->-0.0979775,y2->1.48152,z2->-0.0457618,x3->0.856444,y3->0.5336,z3->0.0690312}}

8

0.198762

1.01592

{{x0->0.718894,y0->1.44948,z0->0.362498,x2->0.865718,y2->0.517926,z2->0.025148,x3->-0.0883416,y3->1.47477,z3->-0.0586878},{x0->0.379372,y0->0.519457,z0->1.98219,x2->0.199318,y2->0.178799,z2->1.77166,x3->-2.62939,y3->1.24777,z3->3.37694},{x0->0.379372,y0->0.519457,z0->1.98219,x2->-2.62939,y2->1.24777,z2->3.37694,x3->0.199318,y3->0.178799,z3->1.77166},{x0->0.718894,y0->1.44948,z0->0.362498,x2->-0.0883416,y2->1.47477,z2->-0.0586878,x3->0.865718,y3->0.517926,z3->0.025148}}

9

0.223607

1.00554

{{x0->0.728018,y0->1.44925,z0->0.325706,x2->0.871887,y2->0.506034,z2->-0.0176261,x3->-0.0816467,y3->1.46997,z3->-0.0704842},{x0->0.384102,y0->0.499523,z0->1.9889,x2->-2.40708,y2->1.2431,z2->3.24618,x3->0.19855,y3->0.164047,z3->1.77538},{x0->0.384102,y0->0.499523,z0->1.9889,x2->0.19855,y2->0.164047,z2->1.77538,x3->-2.40708,y3->1.2431,z3->3.24618},{x0->0.728018,y0->1.44925,z0->0.325706,x2->-0.0816467,y2->1.46997,z2->-0.0704842,x3->0.871887,y3->0.506034,z3->-0.0176261}}

10

0.248452

0.993808

{{x0->0.73548,y0->1.44909,z0->0.291635,x2->0.875374,y2->0.497513,z2->-0.0588493,x3->-0.0772591,y3->1.46672,z3->-0.0806411},{x0->0.388056,y0->0.479863,z0->1.99542,x2->-2.20641,y2->1.23365,z2->3.1276,x3->0.197606,y3->0.149366,z3->1.77903},{x0->0.388056,y0->0.479863,z0->1.99542,x2->0.197606,y2->0.149366,z2->1.77903,x3->-2.20641,y3->1.23365,z3->3.1276},{x0->0.73548,y0->1.44909,z0->0.291635,x2->-0.0772591,y2->1.46672,z2->-0.0806411,x3->0.875374,y3->0.497513,z3->-0.0588493}}

11

0.273297

0.980678

{{x0->0.741503,y0->1.44895,z0->0.26062,x2->0.876579,y2->0.491958,z2->-0.0981796,x3->-0.0746826,y3->1.46473,z3->-0.0887314},{x0->0.391305,y0->0.46052,z0->2.00174,x2->-2.02456,y2->1.22055,z2->3.01973,x3->0.196487,y3->0.134752,z3->1.78262},{x0->0.391305,y0->0.46052,z0->2.00174,x2->0.196487,y2->0.134752,z2->1.78262,x3->-2.02456,y3->1.22055,z3->3.01973},{x0->0.741503,y0->1.44895,z0->0.26062,x2->-0.0746826,y2->1.46473,z2->-0.0887314,x3->0.876579,y3->0.491958,z3->-0.0981796}}

12

0.298142

0.966092

{{x0->0.746292,y0->1.44883,z0->0.232946,x2->0.875862,y2->0.488995,z2->-0.135359,x3->-0.0735258,y3->1.46376,z3->-0.0943854},{x0->0.393915,y0->0.441526,z0->2.00784,x2->-1.85897,y2->1.20466,z2->2.92121,x3->0.195191,y3->0.120192,z3->1.78615},{x0->0.393915,y0->0.441526,z0->2.00784,x2->0.195191,y2->0.120192,z2->1.78615,x3->-1.85897,y3->1.20466,z3->2.92121},{x0->0.746292,y0->1.44883,z0->0.232946,x2->-0.0735258,y2->1.46376,z2->-0.0943854,x3->0.875862,y3->0.488995,z3->-0.135359}}

13

0.322988

0.949984

{{x0->0.750038,y0->1.44871,z0->0.208867,x2->0.873544,y2->0.488291,z2->-0.170194,x3->-0.0734768,y3->1.46363,z3->-0.0972681},{x0->0.395942,y0->0.422908,z0->2.01371,x2->-1.70741,y2->1.18658,z2->2.83085,x3->0.193712,y3->0.105672,z3->1.78964},{x0->0.395942,y0->0.422908,z0->2.01371,x2->0.193712,y2->0.105672,z2->1.78964,x3->-1.70741,y3->1.18658,z3->2.83085},{x0->0.750038,y0->1.44871,z0->0.208867,x2->-0.0734768,y2->1.46363,z2->-0.0972681,x3->0.873544,y3->0.488291,z3->-0.170194}}

14

0.347833

0.932275

{{x0->0.397439,y0->0.404685,z0->2.01936,x2->-1.56794,y2->1.16677,z2->2.74756,x3->0.192042,y3->0.091172,z3->1.7931},{x0->0.752908,y0->1.44859,z0->0.188618,x2->0.869906,y2->0.489569,z2->-0.202539,x3->-0.0742853,y3->1.46419,z3->-0.0970586},{x0->0.397439,y0->0.404685,z0->2.01936,x2->0.192042,y2->0.091172,z2->1.7931,x3->-1.56794,y3->1.16677,z3->2.74756},{x0->0.752908,y0->1.44859,z0->0.188618,x2->-0.0742853,y2->1.46419,z2->-0.0970586,x3->0.869906,y3->0.489569,z3->-0.202539}}

15

0.372678

0.912871

{{x0->0.39845,y0->0.386875,z0->2.02477,x2->-1.43888,y2->1.14554,z2->2.67039,x3->0.190165,y3->0.0766674,z3->1.79655},{x0->0.75505,y0->1.44846,z0->0.172442,x2->0.865183,y2->0.492602,z2->-0.232278,x3->-0.0757484,y3->1.46531,z3->-0.0934305},{x0->0.39845,y0->0.386875,z0->2.02477,x2->0.190165,y2->0.0766674,z2->1.79655,x3->-1.43888,y3->1.14554,z3->2.67039},{x0->0.75505,y0->1.44846,z0->0.172442,x2->-0.0757484,y2->1.46531,z2->-0.0934305,x3->0.865183,y3->0.492602,z3->-0.232278}}

16

0.397523

0.891662

{{x0->0.399015,y0->0.369494,z0->2.02995,x2->-1.31877,y2->1.12311,z2->2.59852,x3->0.188062,y3->0.0621307,z3->1.80001},{x0->0.399015,y0->0.369494,z0->2.02995,x2->0.188062,y2->0.0621307,z2->1.80001,x3->-1.31877,y3->1.12311,z3->2.59852},{x0->0.75659,y0->1.44831,z0->0.160605,x2->0.859575,y2->0.497221,z2->-0.259311,x3->-0.0777,y3->1.46687,z3->-0.086031},{x0->0.75659,y0->1.44831,z0->0.160605,x2->-0.0777,y2->1.46687,z2->-0.086031,x3->0.859575,y3->0.497221,z3->-0.259311}}

17

0.422368

0.868517

{{x0->0.399168,y0->0.352562,z0->2.0349,x2->0.185705,y2->0.04753,z2->1.80351,x3->-1.20637,y3->1.09963,z3->2.53122},{x0->0.399168,y0->0.352562,z0->2.0349,x2->-1.20637,y2->1.09963,z2->2.53122,x3->0.185705,y3->0.04753,z3->1.80351},{x0->0.757634,y0->1.44813,z0->0.153417,x2->0.853238,y2->0.503317,z2->-0.283535,x3->-0.0800037,y3->1.46875,z3->-0.0744591},{x0->0.757634,y0->1.44813,z0->0.153417,x2->-0.0800037,y2->1.46875,z2->-0.0744591,x3->0.853238,y3->0.503317,z3->-0.283535}}

18

0.447214

0.843274

{{x0->0.398937,y0->0.336103,z0->2.03963,x2->0.183056,y2->0.0328296,z2->1.8071,x3->-1.10057,y3->1.07518,z3->2.46785},{x0->0.398937,y0->0.336103,z0->2.03963,x2->-1.10057,y2->1.07518,z2->2.46785,x3->0.183056,y3->0.0328296,z3->1.8071},{x0->0.758267,y0->1.44786,z0->0.151259,x2->0.84629,y2->0.51084,z2->-0.304833,x3->-0.0825478,y3->1.47085,z3->-0.0582376},{x0->0.758267,y0->1.44786,z0->0.151259,x2->-0.0825478,y2->1.47085,z2->-0.0582376,x3->0.84629,y3->0.51084,z3->-0.304833}}

19

0.472059

0.81574

{{x0->0.398346,y0->0.320149,z0->2.04413,x2->0.180069,y2->0.0179895,z2->1.81082,x3->-1.00043,y3->1.0498,z3->2.40781},{x0->0.398346,y0->0.320149,z0->2.04413,x2->-1.00043,y2->1.0498,z2->2.40781,x3->0.180069,y3->0.0179895,z3->1.81082},{x0->0.758553,y0->1.44747,z0->0.154614,x2->0.838804,y2->0.519817,z2->-0.323054,x3->-0.0852421,y3->1.47303,z3->-0.0367781},{x0->0.758553,y0->1.44747,z0->0.154614,x2->-0.0852421,y2->1.47303,z2->-0.0367781,x3->0.838804,y3->0.519817,z3->-0.323054}}

20

0.496904

0.785674

{{x0->0.397418,y0->0.30475,z0->2.04841,x2->0.176677,y2->0.0029654,z2->1.81474,x3->-0.905098,y3->1.0235,z3->2.35059},{x0->0.397418,y0->0.30475,z0->2.04841,x2->-0.905098,y2->1.0235,z2->2.35059,x3->0.176677,y3->0.0029654,z3->1.81474},{x0->0.758532,y0->1.44686,z0->0.164109,x2->0.8308,y2->0.530361,z2->-0.337988,x3->-0.0880173,y3->1.47512,z3->-0.0093319},{x0->0.758532,y0->1.44686,z0->0.164109,x2->-0.0880173,y2->1.47512,z2->-0.0093319,x3->0.8308,y3->0.530361,z3->-0.337988}}

21

0.521749

0.752773

{{x0->0.39617,y0->0.289976,z0->2.05247,x2->0.172792,y2->-0.0122904,z2->1.81894,x3->-0.81379,y3->0.996237,z3->2.29569},{x0->0.39617,y0->0.289976,z0->2.05247,x2->-0.81379,y2->0.996237,z2->2.29569,x3->0.172792,y3->-0.0122904,z3->1.81894},{x0->0.758218,y0->1.44588,z0->0.180576,x2->0.822226,y2->0.542707,z2->-0.349333,x3->-0.0908273,y3->1.47694,z3->0.0250828},{x0->0.758218,y0->1.44588,z0->0.180576,x2->-0.0908273,y2->1.47694,z2->0.0250828,x3->0.822226,y3->0.542707,z3->-0.349333}}

22

0.546594

0.716645

{{x0->0.394618,y0->0.275939,z0->2.05629,x2->0.168292,y2->-0.0278278,z2->1.82354,x3->-0.725786,y3->0.967953,z3->2.24261},{x0->0.394618,y0->0.275939,z0->2.05629,x2->-0.725786,y2->0.967953,z2->2.24261,x3->0.168292,y3->-0.0278278,z3->1.82354},{x0->0.757582,y0->1.44431,z0->0.205143,x2->0.812933,y2->0.557258,z2->-0.356641,x3->-0.0936557,y3->1.47818,z3->0.0677938},{x0->0.757582,y0->1.44431,z0->0.205143,x2->-0.0936557,y2->1.47818,z2->0.0677938,x3->0.812933,y3->0.557258,z3->-0.356641}}

23

0.57144

0.676775

{{x0->0.392779,y0->0.262812,z0->2.05985,x2->0.163004,y2->-0.0436931,z2->1.82871,x3->-0.640377,y3->0.938562,z3->2.19082},{x0->0.392779,y0->0.262812,z0->2.05985,x2->-0.640377,y2->0.938562,z2->2.19082,x3->0.163004,y3->-0.0436931,z3->1.82871},{x0->0.756541,y0->1.4418,z0->0.239384,x2->0.802609,y2->0.574686,z2->-0.359217,x3->-0.0965297,y3->1.47844,z3->0.120662},{x0->0.756541,y0->1.4418,z0->0.239384,x2->-0.0965297,y2->1.47844,z2->0.120662,x3->0.802609,y3->0.574686,z3->-0.359217}}

24

0.596285

0.632456

{{x0->0.390676,y0->0.250881,z0->2.06313,x2->0.156671,y2->-0.0599157,z2->1.8347,x3->-0.556826,y3->0.907965,z3->2.13974},{x0->0.390676,y0->0.250881,z0->2.06313,x2->-0.556826,y2->0.907965,z2->2.13974,x3->0.156671,y3->-0.0599157,z3->1.8347},{x0->0.754915,y0->1.43777,z0->0.285576,x2->0.790659,y2->0.596104,z2->-0.355929,x3->-0.0995509,y3->1.47707,z3->0.186411},{x0->0.754915,y0->1.43777,z0->0.285576,x2->-0.0995509,y2->1.47707,z2->0.186411,x3->0.790659,y3->0.596104,z3->-0.355929}}

25

0.62113

0.582672

{{x0->0.38834,y0->0.240648,z0->2.06604,x2->0.148892,y2->-0.0764711,z2->1.84194,x3->-0.474296,y3->0.876071,z3->2.08858},{x0->0.38834,y0->0.240648,z0->2.06604,x2->-0.474296,y2->0.876071,z2->2.08858,x3->0.148892,y3->-0.0764711,z3->1.84194},{x0->0.752353,y0->1.43124,z0->0.347159,x2->-0.102961,y2->1.47298,z2->0.269265,x3->0.775948,y3->0.623429,z3->-0.344793},{x0->0.752353,y0->1.43124,z0->0.347159,x2->0.775948,y2->0.623429,z2->-0.344793,x3->-0.102961,y3->1.47298,z3->0.269265}}

26

0.645975

0.525874

{{x0->0.385839,y0->0.233066,z0->2.06843,x2->0.138995,y2->-0.0931745,z2->1.85112,x3->-0.391716,y3->0.842876,z3->2.03617},{x0->0.385839,y0->0.233066,z0->2.06843,x2->-0.391716,y2->0.842876,z2->2.03617,x3->0.138995,y3->-0.0931745,z3->1.85112},{x0->0.748156,y0->1.4204,z0->0.429705,x2->-0.1073,y2->1.46417,z2->0.376305,x3->0.756176,y3->0.660167,z3->-0.321958},{x0->0.748156,y0->1.4204,z0->0.429705,x2->0.756176,y2->0.660167,z2->-0.321958,x3->-0.1073,y3->1.46417,z3->0.376305}}

27

0.67082

0.459468

{{x0->0.383337,y0->0.230207,z0->2.06989,x2->0.12572,y2->-0.10933,z2->1.86366,x3->-0.307455,y3->0.808725,z3->1.98034},{x0->0.383337,y0->0.230207,z0->2.06989,x2->-0.307455,y2->0.808725,z2->1.98034,x3->0.12572,y3->-0.10933,z3->1.86366},{x0->0.740818,y0->1.40157,z0->0.543165,x2->-0.11385,y2->1.44639,z2->0.520885,x3->0.726155,y3->0.713337,z3->-0.278709},{x0->0.740818,y0->1.40157,z0->0.543165,x2->0.726155,y2->0.713337,z2->-0.278709,x3->-0.11385,y3->1.44639,z3->0.520885}}

28

0.695666

0.378431

{{x0->0.381348,y0->0.237658,z0->2.06922,x2->0.106234,y2->-0.122254,z2->1.88286,x3->-0.218365,y3->0.775442,z3->1.91594},{x0->0.381348,y0->0.237658,z0->2.06922,x2->-0.218365,y2->0.775442,z2->1.91594,x3->0.106234,y3->-0.122254,z3->1.88286},{x0->0.726508,y0->1.36582,z0->0.708478,x2->-0.126166,y2->1.4083,z2->0.733519,x3->0.671873,y3->0.798864,z3->-0.190111},{x0->0.726508,y0->1.36582,z0->0.708478,x2->0.671873,y2->0.798864,z2->-0.190111,x3->-0.126166,y3->1.4083,z3->0.733519}}

29

0.720511

0.269888

{{x0->0.382569,y0->0.278823,z0->2.06019,x2->0.0712543,y2->-0.117004,z2->1.92031,x3->-0.115107,y3->0.754072,z3->1.82296},{x0->0.691266,y0->1.28141,z0->0.987048,x2->-0.157484,y2->1.30095,z2->1.11579,x3->0.541792,y3->0.956619,z3->0.0509191},{x0->0.382569,y0->0.278823,z0->2.06019,x2->-0.115107,y2->0.754072,z2->1.82296,x3->0.0712543,y3->-0.117004,z3->1.92031},{x0->0.691266,y0->1.28141,z0->0.987048,x2->0.541792,y2->0.956619,z2->0.0509191,x3->-0.157484,y3->1.30095,z3->1.11579}}

30

0.745356

0.

{}

从表面上看，$k=30$ 时无解，此时 $x_1=\sqrt{5}/3$，$y_1=0$，由45#的方法，可得
\[
0.4112136430677168\leq x_0 \leq 1.505416051932103
\]
取 $x_0=1$，可得其中一组 $y_0$、$z_0$ 值
\[
y_0=\sqrt{\frac{2}{5} \left(9 \sqrt{5}-17\right)},z_0=3-\sqrt{5}
\]
再运行
NSolve[{a x0 x + b y0 y + c z0 z ==
   1, (x1 - x0) (x - x0) + (y1 - y0) (y - y0) + (z1 - z0) (z - z0) ==
   0, a x^2 + b y^2 + c z^2 == 1}, {x, y, z}, Reals]
得结果
{{x -> 0.896466, y -> 0.412962, z -> -0.666574}, {x -> 0.131804,
  y -> 1.30162, z -> 0.73625
取结果分别为 $(x_2,y_2,z_2)$、 $(x_3,y_3,z_3)$，计算验证 $(x_2-x_0)(x_3-x_0)+(y_2-y_0)(y_3-y_0)+(z_2-z_0)(z_3-z_0)$ 的值确实为0，由此可知，并非没有解，而确实是有无数解。

hejoseph

mathe 发表于 2018-12-1 19:32
应该这样推广。给定一个二次曲面$\Gamma$, 其上三根切线$l_1,l_2,l_3$相交于点$P$,对应切点分别为$A,B,C$， ...

对应这个问题：椭圆两切线交于点 $P$ 且夹角为非直角的定值，则点 $P$ 的轨迹是一条四次曲线。对于三维情形，应该更复杂，不会再是二次曲面了。