过三定点的抛物线之焦点轨迹是五次曲线

1+1=？ · 2025-5-11 02:12

Last edited by 1+1=？ 2025-5-26 10:40求证过三定点的抛物线焦点轨迹为五次曲线，且求三个定点在什么条件下，焦点轨迹退化为非五次曲线

hbghlyj · 2025-5-22 06:30

1906年的论文A Problem in Conics. by RF DAVIS — On the other hand the analogous locus of the focus of a parabola passing through three given fixed points is…\[
(2 s-3 a) r^4-2 s(s-a)^2 r^2+a(s-a)^4=0
\]
where $s=x+y, r^2=x^2+y^2$. The result is therefore a quintic having the lines $x=\frac{a}{2}, y=\frac{a}{2}, x+y-\frac{3 a}{2}=0$ as asymptotes.

1+1=？ · 2025-5-22 11:45

hbghlyj 发表于 2025-5-22 06:30
1906年的论文A Problem in Conics. by RF DAVIS — On the other hand the analogous locus of the focus o ...

1906年的论文都找到了

creasson · 2025-5-22 14:14

过A,B,C三点的抛物线，焦点Z的轨迹是
\[Z = B + \frac{{{{\left( { - B + C - A{s^2} + B{s^2}} \right)}^2}}}{{4s\left( {1 + s} \right)\left( {B - C - As + Bs} \right)}}\]
其中 s 是自由参数。这可以从如下两个结论得到：

转化为直角坐标是5次方程。

又由marden定理

与三角形ABC三边相切的抛物线，焦点Z的轨迹是
\[Z = B - \frac{{\left( { - A + B} \right)\left( {B - C} \right)\left( { - 1 + s} \right)}}{{ - B + C - As + Bs}}\]
转化为直角坐标是2次方程。

hejoseph · 2025-5-23 09:10

与定三角形三边所在直线相切的抛物线族的焦点轨迹显然就是三角形的外接圆。

hejoseph · 2025-5-23 10:44

Last edited by hejoseph 2025-5-23 14:39如果抛物线主轴的方向是重心坐标为 $\alpha:\beta:\gamma$ 的无穷远点，则 $\triangle ABC$ 外接抛物线焦点的重心坐标为 $x_1:x_2:x_3$，其中
\begin{align*}
x_1&=\alpha\left(\beta\gamma(\alpha^2+\beta\gamma)a^2+\alpha^3(\gamma b^2+\beta c^2)\right)\\
x_2&=\beta\left(\gamma\alpha(\beta^2+\gamma\alpha)b^2+\beta^3(\alpha c^2+\gamma a^2)\right)\\
x_3&=\gamma\left(\alpha\beta(\gamma^2+\alpha\beta)c^2+\gamma^3(\beta a^2+\alpha b^2)\right)
\end{align*}
利用上面的结论结合 $\alpha+\beta+\gamma=0$ 消去 $\alpha$、$\beta$、$\gamma$ 就能得到 $\triangle ABC$ 外接抛物线焦点轨迹的重心坐标方程，得到的是一个五次多项式。
Mathematica运行下面命令后可以得到消元结果：

Eliminate[{x1==α (β γ (α^2+β γ) a^2+α^3 (γ b^2+β c^2)),x2==β (γ α (β^2+γ α ) b^2+β^3 (α c^2+γ a^2)),x3==γ (α β (γ^2+α β) c^2+γ^3 (β a^2+α b^2)),α+β+γ==0},{α,β,γ}]

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消元后的结果太复杂，直接贴出来：

a^8 x1^5 - 4 a^6 b^2 x1^5 + 6 a^4 b^4 x1^5 -
4 a^2 b^6 x1^5 + b^8 x1^5 - 4 a^6 c^2 x1^5 +
4 a^4 b^2 c^2 x1^5 + 4 a^2 b^4 c^2 x1^5 -
4 b^6 c^2 x1^5 + 6 a^4 c^4 x1^5 + 4 a^2 b^2 c^4 x1^5 +
6 b^4 c^4 x1^5 - 4 a^2 c^6 x1^5 - 4 b^2 c^6 x1^5 +
c^8 x1^5 + 5 a^8 x1^4 x2 - 20 a^6 b^2 x1^4 x2 +
30 a^4 b^4 x1^4 x2 - 20 a^2 b^6 x1^4 x2 + 5 b^8 x1^4 x2 -
36 a^6 c^2 x1^4 x2 + 100 a^4 b^2 c^2 x1^4 x2 -
92 a^2 b^4 c^2 x1^4 x2 + 28 b^6 c^2 x1^4 x2 +
78 a^4 c^4 x1^4 x2 - 76 a^2 b^2 c^4 x1^4 x2 -
50 b^4 c^4 x1^4 x2 - 68 a^2 c^6 x1^4 x2 -
4 b^2 c^6 x1^4 x2 + 21 c^8 x1^4 x2 + 10 a^8 x1^3 x2^2 -
40 a^6 b^2 x1^3 x2^2 + 60 a^4 b^4 x1^3 x2^2 -
40 a^2 b^6 x1^3 x2^2 + 10 b^8 x1^3 x2^2 -
24 a^6 c^2 x1^3 x2^2 + 88 a^4 b^2 c^2 x1^3 x2^2 -
104 a^2 b^4 c^2 x1^3 x2^2 + 40 b^6 c^2 x1^3 x2^2 +
76 a^4 c^4 x1^3 x2^2 + 8 a^2 b^2 c^4 x1^3 x2^2 -
52 b^4 c^4 x1^3 x2^2 - 120 a^2 c^6 x1^3 x2^2 -
56 b^2 c^6 x1^3 x2^2 + 58 c^8 x1^3 x2^2 +
10 a^8 x1^2 x2^3 - 40 a^6 b^2 x1^2 x2^3 +
60 a^4 b^4 x1^2 x2^3 - 40 a^2 b^6 x1^2 x2^3 +
10 b^8 x1^2 x2^3 + 40 a^6 c^2 x1^2 x2^3 -
104 a^4 b^2 c^2 x1^2 x2^3 + 88 a^2 b^4 c^2 x1^2 x2^3 -
24 b^6 c^2 x1^2 x2^3 - 52 a^4 c^4 x1^2 x2^3 +
8 a^2 b^2 c^4 x1^2 x2^3 + 76 b^4 c^4 x1^2 x2^3 -
56 a^2 c^6 x1^2 x2^3 - 120 b^2 c^6 x1^2 x2^3 +
58 c^8 x1^2 x2^3 + 5 a^8 x1 x2^4 - 20 a^6 b^2 x1 x2^4 +
30 a^4 b^4 x1 x2^4 - 20 a^2 b^6 x1 x2^4 + 5 b^8 x1 x2^4 +
28 a^6 c^2 x1 x2^4 - 92 a^4 b^2 c^2 x1 x2^4 +
100 a^2 b^4 c^2 x1 x2^4 - 36 b^6 c^2 x1 x2^4 -
50 a^4 c^4 x1 x2^4 - 76 a^2 b^2 c^4 x1 x2^4 +
78 b^4 c^4 x1 x2^4 - 4 a^2 c^6 x1 x2^4 -
68 b^2 c^6 x1 x2^4 + 21 c^8 x1 x2^4 + a^8 x2^5 -
4 a^6 b^2 x2^5 + 6 a^4 b^4 x2^5 - 4 a^2 b^6 x2^5 +
b^8 x2^5 - 4 a^6 c^2 x2^5 + 4 a^4 b^2 c^2 x2^5 +
4 a^2 b^4 c^2 x2^5 - 4 b^6 c^2 x2^5 + 6 a^4 c^4 x2^5 +
4 a^2 b^2 c^4 x2^5 + 6 b^4 c^4 x2^5 - 4 a^2 c^6 x2^5 -
4 b^2 c^6 x2^5 + c^8 x2^5 + 5 a^8 x1^4 x3 -
36 a^6 b^2 x1^4 x3 + 78 a^4 b^4 x1^4 x3 -
68 a^2 b^6 x1^4 x3 + 21 b^8 x1^4 x3 -
20 a^6 c^2 x1^4 x3 + 100 a^4 b^2 c^2 x1^4 x3 -
76 a^2 b^4 c^2 x1^4 x3 - 4 b^6 c^2 x1^4 x3 +
30 a^4 c^4 x1^4 x3 - 92 a^2 b^2 c^4 x1^4 x3 -
50 b^4 c^4 x1^4 x3 - 20 a^2 c^6 x1^4 x3 +
28 b^2 c^6 x1^4 x3 + 5 c^8 x1^4 x3 - 28 a^8 x1^3 x2 x3 +
80 a^6 b^2 x1^3 x2 x3 - 72 a^4 b^4 x1^3 x2 x3 +
16 a^2 b^6 x1^3 x2 x3 + 4 b^8 x1^3 x2 x3 +
80 a^6 c^2 x1^3 x2 x3 - 304 a^4 b^2 c^2 x1^3 x2 x3 +
240 a^2 b^4 c^2 x1^3 x2 x3 - 16 b^6 c^2 x1^3 x2 x3 -
72 a^4 c^4 x1^3 x2 x3 + 240 a^2 b^2 c^4 x1^3 x2 x3 +
24 b^4 c^4 x1^3 x2 x3 + 16 a^2 c^6 x1^3 x2 x3 -
16 b^2 c^6 x1^3 x2 x3 + 4 c^8 x1^3 x2 x3 -
50 a^8 x1^2 x2^2 x3 + 200 a^6 b^2 x1^2 x2^2 x3 -
300 a^4 b^4 x1^2 x2^2 x3 + 200 a^2 b^6 x1^2 x2^2 x3 -
50 b^8 x1^2 x2^2 x3 + 88 a^6 c^2 x1^2 x2^2 x3 -
88 a^4 b^2 c^2 x1^2 x2^2 x3 -
88 a^2 b^4 c^2 x1^2 x2^2 x3 + 88 b^6 c^2 x1^2 x2^2 x3 -
28 a^4 c^4 x1^2 x2^2 x3 - 104 a^2 b^2 c^4 x1^2 x2^2 x3 -
28 b^4 c^4 x1^2 x2^2 x3 - 8 a^2 c^6 x1^2 x2^2 x3 -
8 b^2 c^6 x1^2 x2^2 x3 - 2 c^8 x1^2 x2^2 x3 +
4 a^8 x1 x2^3 x3 + 16 a^6 b^2 x1 x2^3 x3 -
72 a^4 b^4 x1 x2^3 x3 + 80 a^2 b^6 x1 x2^3 x3 -
28 b^8 x1 x2^3 x3 - 16 a^6 c^2 x1 x2^3 x3 +
240 a^4 b^2 c^2 x1 x2^3 x3 -
304 a^2 b^4 c^2 x1 x2^3 x3 + 80 b^6 c^2 x1 x2^3 x3 +
24 a^4 c^4 x1 x2^3 x3 + 240 a^2 b^2 c^4 x1 x2^3 x3 -
72 b^4 c^4 x1 x2^3 x3 - 16 a^2 c^6 x1 x2^3 x3 +
16 b^2 c^6 x1 x2^3 x3 + 4 c^8 x1 x2^3 x3 +
21 a^8 x2^4 x3 - 68 a^6 b^2 x2^4 x3 +
78 a^4 b^4 x2^4 x3 - 36 a^2 b^6 x2^4 x3 + 5 b^8 x2^4 x3 -
4 a^6 c^2 x2^4 x3 - 76 a^4 b^2 c^2 x2^4 x3 +
100 a^2 b^4 c^2 x2^4 x3 - 20 b^6 c^2 x2^4 x3 -
50 a^4 c^4 x2^4 x3 - 92 a^2 b^2 c^4 x2^4 x3 +
30 b^4 c^4 x2^4 x3 + 28 a^2 c^6 x2^4 x3 -
20 b^2 c^6 x2^4 x3 + 5 c^8 x2^4 x3 + 10 a^8 x1^3 x3^2 -
24 a^6 b^2 x1^3 x3^2 + 76 a^4 b^4 x1^3 x3^2 -
120 a^2 b^6 x1^3 x3^2 + 58 b^8 x1^3 x3^2 -
40 a^6 c^2 x1^3 x3^2 + 88 a^4 b^2 c^2 x1^3 x3^2 +
8 a^2 b^4 c^2 x1^3 x3^2 - 56 b^6 c^2 x1^3 x3^2 +
60 a^4 c^4 x1^3 x3^2 - 104 a^2 b^2 c^4 x1^3 x3^2 -
52 b^4 c^4 x1^3 x3^2 - 40 a^2 c^6 x1^3 x3^2 +
40 b^2 c^6 x1^3 x3^2 + 10 c^8 x1^3 x3^2 -
50 a^8 x1^2 x2 x3^2 + 88 a^6 b^2 x1^2 x2 x3^2 -
28 a^4 b^4 x1^2 x2 x3^2 - 8 a^2 b^6 x1^2 x2 x3^2 -
2 b^8 x1^2 x2 x3^2 + 200 a^6 c^2 x1^2 x2 x3^2 -
88 a^4 b^2 c^2 x1^2 x2 x3^2 -
104 a^2 b^4 c^2 x1^2 x2 x3^2 - 8 b^6 c^2 x1^2 x2 x3^2 -
300 a^4 c^4 x1^2 x2 x3^2 - 88 a^2 b^2 c^4 x1^2 x2 x3^2 -
28 b^4 c^4 x1^2 x2 x3^2 + 200 a^2 c^6 x1^2 x2 x3^2 +
88 b^2 c^6 x1^2 x2 x3^2 - 50 c^8 x1^2 x2 x3^2 -
2 a^8 x1 x2^2 x3^2 - 8 a^6 b^2 x1 x2^2 x3^2 -
28 a^4 b^4 x1 x2^2 x3^2 + 88 a^2 b^6 x1 x2^2 x3^2 -
50 b^8 x1 x2^2 x3^2 - 8 a^6 c^2 x1 x2^2 x3^2 -
104 a^4 b^2 c^2 x1 x2^2 x3^2 -
88 a^2 b^4 c^2 x1 x2^2 x3^2 + 200 b^6 c^2 x1 x2^2 x3^2 -
28 a^4 c^4 x1 x2^2 x3^2 - 88 a^2 b^2 c^4 x1 x2^2 x3^2 -
300 b^4 c^4 x1 x2^2 x3^2 + 88 a^2 c^6 x1 x2^2 x3^2 +
200 b^2 c^6 x1 x2^2 x3^2 - 50 c^8 x1 x2^2 x3^2 +
58 a^8 x2^3 x3^2 - 120 a^6 b^2 x2^3 x3^2 +
76 a^4 b^4 x2^3 x3^2 - 24 a^2 b^6 x2^3 x3^2 +
10 b^8 x2^3 x3^2 - 56 a^6 c^2 x2^3 x3^2 +
8 a^4 b^2 c^2 x2^3 x3^2 + 88 a^2 b^4 c^2 x2^3 x3^2 -
40 b^6 c^2 x2^3 x3^2 - 52 a^4 c^4 x2^3 x3^2 -
104 a^2 b^2 c^4 x2^3 x3^2 + 60 b^4 c^4 x2^3 x3^2 +
40 a^2 c^6 x2^3 x3^2 - 40 b^2 c^6 x2^3 x3^2 +
10 c^8 x2^3 x3^2 + 10 a^8 x1^2 x3^3 +
40 a^6 b^2 x1^2 x3^3 - 52 a^4 b^4 x1^2 x3^3 -
56 a^2 b^6 x1^2 x3^3 + 58 b^8 x1^2 x3^3 -
40 a^6 c^2 x1^2 x3^3 - 104 a^4 b^2 c^2 x1^2 x3^3 +
8 a^2 b^4 c^2 x1^2 x3^3 - 120 b^6 c^2 x1^2 x3^3 +
60 a^4 c^4 x1^2 x3^3 + 88 a^2 b^2 c^4 x1^2 x3^3 +
76 b^4 c^4 x1^2 x3^3 - 40 a^2 c^6 x1^2 x3^3 -
24 b^2 c^6 x1^2 x3^3 + 10 c^8 x1^2 x3^3 +
4 a^8 x1 x2 x3^3 - 16 a^6 b^2 x1 x2 x3^3 +
24 a^4 b^4 x1 x2 x3^3 - 16 a^2 b^6 x1 x2 x3^3 +
4 b^8 x1 x2 x3^3 + 16 a^6 c^2 x1 x2 x3^3 +
240 a^4 b^2 c^2 x1 x2 x3^3 +
240 a^2 b^4 c^2 x1 x2 x3^3 + 16 b^6 c^2 x1 x2 x3^3 -
72 a^4 c^4 x1 x2 x3^3 - 304 a^2 b^2 c^4 x1 x2 x3^3 -
72 b^4 c^4 x1 x2 x3^3 + 80 a^2 c^6 x1 x2 x3^3 +
80 b^2 c^6 x1 x2 x3^3 - 28 c^8 x1 x2 x3^3 +
58 a^8 x2^2 x3^3 - 56 a^6 b^2 x2^2 x3^3 -
52 a^4 b^4 x2^2 x3^3 + 40 a^2 b^6 x2^2 x3^3 +
10 b^8 x2^2 x3^3 - 120 a^6 c^2 x2^2 x3^3 +
8 a^4 b^2 c^2 x2^2 x3^3 - 104 a^2 b^4 c^2 x2^2 x3^3 -
40 b^6 c^2 x2^2 x3^3 + 76 a^4 c^4 x2^2 x3^3 +
88 a^2 b^2 c^4 x2^2 x3^3 + 60 b^4 c^4 x2^2 x3^3 -
24 a^2 c^6 x2^2 x3^3 - 40 b^2 c^6 x2^2 x3^3 +
10 c^8 x2^2 x3^3 + 5 a^8 x1 x3^4 + 28 a^6 b^2 x1 x3^4 -
50 a^4 b^4 x1 x3^4 - 4 a^2 b^6 x1 x3^4 + 21 b^8 x1 x3^4 -
20 a^6 c^2 x1 x3^4 - 92 a^4 b^2 c^2 x1 x3^4 -
76 a^2 b^4 c^2 x1 x3^4 - 68 b^6 c^2 x1 x3^4 +
30 a^4 c^4 x1 x3^4 + 100 a^2 b^2 c^4 x1 x3^4 +
78 b^4 c^4 x1 x3^4 - 20 a^2 c^6 x1 x3^4 -
36 b^2 c^6 x1 x3^4 + 5 c^8 x1 x3^4 + 21 a^8 x2 x3^4 -
4 a^6 b^2 x2 x3^4 - 50 a^4 b^4 x2 x3^4 +
28 a^2 b^6 x2 x3^4 + 5 b^8 x2 x3^4 - 68 a^6 c^2 x2 x3^4 -
76 a^4 b^2 c^2 x2 x3^4 - 92 a^2 b^4 c^2 x2 x3^4 -
20 b^6 c^2 x2 x3^4 + 78 a^4 c^4 x2 x3^4 +
100 a^2 b^2 c^4 x2 x3^4 + 30 b^4 c^4 x2 x3^4 -
36 a^2 c^6 x2 x3^4 - 20 b^2 c^6 x2 x3^4 + 5 c^8 x2 x3^4 +
a^8 x3^5 - 4 a^6 b^2 x3^5 + 6 a^4 b^4 x3^5 -
4 a^2 b^6 x3^5 + b^8 x3^5 - 4 a^6 c^2 x3^5 +
4 a^4 b^2 c^2 x3^5 + 4 a^2 b^4 c^2 x3^5 -
4 b^6 c^2 x3^5 + 6 a^4 c^4 x3^5 + 4 a^2 b^2 c^4 x3^5 +
6 b^4 c^4 x3^5 - 4 a^2 c^6 x3^5 - 4 b^2 c^6 x3^5 + c^8 x3^5

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[几何] 过三定点的抛物线之焦点轨迹是五次曲线

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