指定区间内的求解非线性方程所有零点的数值方法

数学星空

或许此曲线与下面结论有关联：

对于椭圆$x^2/a^2+y^2/b^2=1$内弦长L的线段AB，且P点分$AP=m*PB$,则$P(x,y)$满足下列方程：
$(m+1)^4*(a^2*b^2-a^2*y^2-b^2*x^2)^2*(a^8*b^4*m^4-2*a^8*b^2*m^4*y^2+a^8*m^4*y^4-2*a^6*b^6*m^4-2*a^6*b^4*m^4*x^2+4*a^6*b^4*m^4*y^2+2*a^6*b^2*m^4*x^2*y^2-2*a^6*b^2*m^4*y^4+$
$a^4*b^8*m^4+4*a^4*b^6*m^4*x^2-2*a^4*b^6*m^4*y^2+a^4*b^4*m^4*x^4-4*a^4*b^4*m^4*x^2*y^2+a^4*b^4*m^4*y^4-2*a^2*b^8*m^4*x^2-2*a^2*b^6*m^4*x^4+2*a^2*b^6*m^4*x^2*y^2+b^8*m^4*x^4-$
$4*a^8*b^4*m^3+4*a^8*m^3*y^4+8*a^6*b^6*m^3+8*a^6*b^4*m^3*x^2-8*a^6*b^4*m^3*y^2-4*a^4*b^8*m^3-8*a^4*b^6*m^3*x^2+8*a^4*b^6*m^3*y^2-4*a^4*b^4*m^3*x^4-$
$4*a^4*b^4*m^3*y^4+4*b^8*m^3*x^4+6*a^8*b^4*m^2+4*a^8*b^2*m^2*y^2+6*a^8*m^2*y^4-12*a^6*b^6*m^2-12*a^6*b^4*m^2*x^2+8*a^6*b^4*m^2*y^2-4*a^6*b^2*m^2*x^2*y^2+4*a^6*b^2*m^2*y^4+$
$6*a^4*b^8*m^2+8*a^4*b^6*m^2*x^2-12*a^4*b^6*m^2*y^2+6*a^4*b^4*m^2*x^4+40*a^4*b^4*m^2*x^2*y^2+6*a^4*b^4*m^2*y^4+4*a^2*b^8*m^2*x^2+4*a^2*b^6*m^2*x^4-4*a^2*b^6*m^2*x^2*y^2+$
$ 6*b^8*m^2*x^4-4*a^8*b^4*m+4*a^8*m*y^4+8*a^6*b^6*m+8*a^6*b^4*m*x^2-8*a^6*b^4*m*y^2-4*a^4*b^8*m-8*a^4*b^6*m*x^2+8*a^4*b^6*m*y^2-4*a^4*b^4*m*x^4-4*a^4*b^4*m*y^4+4*b^8*m*x^4+$
$a^8*b^4-2*a^8*b^2*y^2+a^8*y^4-2*a^6*b^6-2*a^6*b^4*x^2+4*a^6*b^4*y^2+2*a^6*b^2*x^2*y^2-2*a^6*b^2*y^4+a^4*b^8+4*a^4*b^6*x^2-2*a^4*b^6*y^2+a^4*b^4*x^4-4*a^4*b^4*x^2*y^2+$
$a^4*b^4*y^4-2*a^2*b^8*x^2-2*a^2*b^6*x^4+2*a^2*b^6*x^2*y^2+b^8*x^4)+8*m^2*a^2*b^2*(m+1)^2*(a^2*b^2-a^2*y^2-b^2*x^2)*(a^6*b^2*m^2*y^2-a^6*m^2*y^4-a^4*b^4*m^2*x^2-$
$a^4*b^4*m^2*y^2+a^4*b^2*m^2*y^4+a^2*b^6*m^2*x^2+a^2*b^4*m^2*x^4-b^6*m^2*x^4-2*a^6*b^2*m*y^2-2*a^6*m*y^4+2*a^4*b^4*m*x^2+2*a^4*b^4*m*y^2-4*a^4*b^2*m*x^2*y^2-$
$2*a^4*b^2*m*y^4-2*a^2*b^6*m*x^2-2*a^2*b^4*m*x^4-4*a^2*b^4*m*x^2*y^2-2*b^6*m*x^4+a^6*b^2*y^2-a^6*y^4-a^4*b^4*x^2-a^4*b^4*y^2+a^4*b^2*y^4+a^2*b^6*x^2+a^2*b^4*x^4-$
$ b^6*x^4)*L^2+16*a^4*b^4*m^4*(a^2*y^2+b^2*x^2)^2*L^4=0$

当P点为AB的中点时即$m=1,L=2*l $时
$ 16*(L^2*a^4*b^2*y^2+L^2*a^2*b^4*x^2-4*a^6*b^2*y^2+4*a^6*y^4+4*a^4*b^2*x^2*y^2-4*a^2*b^6*x^2+4*a^2*b^4*x^2*y^2+4*b^6*x^4)^2=0$
即陈都得到的结果：
http://bbs.emath.ac.cn/forum.php?mod=viewthread&tid=4216&extra=page%3D1&page=6
56# 椭圆定长弦中点轨迹：
$ (1-x^2/a^2-y^2/b^2)*(b^2*x^2/a^2+a^2*y^2/b^2)=(x^2/a^2+y^2/b^2)*l^2$

mathe

设椭圆上两点$(a*cos(t_1),b*sin(t_1)), (a*cos(t_2), b*sin(t_2))$距离是常数$L$
于是两点确定直线方程为$y-{b(sin(t_2)+sin(t_1))}/2=b/a{sin(t_2)-sin(t_1)}/{cos(t_2)-cos(t_1)}(x-{a(cos(t_2)+cos(t_1))}/2)$
可以和差化积简化公式，类似对距离平方和为$L^2$也用和差化积简化，然后可以改参数$t_1,t_2$为参数$u={t_1+t_2}/2,v={t_1-t_2}/2$写出直线的参数方程。

mathe

然后设曲线列为$x/{a(t)}+y/{b(t)}=1$
那么对于曲线上一点，我们需要计算t和$t_2$对应直线的交点坐标然后让$t_2$趋向t即可
得出
${(b(t)x+a(t)y-a(t)b(t)=0),(b(t_2)x+a(t_2)y-a(t_2)b(t_2)=0):}$
$x={a(t)a(t_2)(b(t)-b(t_2))}/{b(t)a(t_2)-b(t_2)a(t)}$
所以$x={a(t)^2b'(t)}/{b'(t)a(t)-b(t)a'(t)}$
所以根据对称性应该还有$y={b(t)^2a'(t)}/{a'(t)b(t)-b'(t)a(t)}$

数学星空

根据楼上mathe提供的计算方案：
我们得到

$y/{{b*cos(v)}/sin(u)}+x/{{a*cos(v)}/cos(u)}=1$

$4*sin(v)^2*(a^2*sin(u)^2+b^2*cos(u)^2)=L^2$

消元并令$sin(u)=t$,得到

$a(t)=b/{a*t}*sqrt(-(-4*a^2*t^2+4*b^2*t^2+L^2-b^2)/(4*a^2*t^2-4*b^2*t^2+b^2))$

$b(t)={a*b}/sqrt(-t^2+1)*sqrt(-(-4*a^2*t^2+4*b^2*t^2+L^2-b^2)/(4*a^2*t^2-4*b^2*t^2+b^2))$

代入$x={a(t)^2b'(t)}/{b'(t)a(t)-b(t)a'(t)}$中有理化后得到

$-256*a^10*t^8*x^2+256*a^8*b^2*t^10+1024*a^8*b^2*t^8*x^2-1024*a^6*b^4*t^10-1536*a^6*b^4*t^8*x^2+1536*a^4*b^6*t^10+1024*a^4*b^6*t^8*x^2-1024*a^2*b^8*t^10-256*a^2*b^8*t^8*x^2+256*b^10*t^10+64*L^2*a^8*t^6*x^2-256*L^2*a^6*b^2*t^8-192*L^2*a^6*b^2*t^6*x^2+768*L^2*a^4*b^4*t^8+192*L^2*a^4*b^4*t^6*x^2-768*L^2*a^2*b^6*t^8-64*L^2*a^2*b^6*t^6*x^2+256*L^2*b^8*t^8-256*a^8*b^2*t^6*x^2+256*a^6*b^4*t^8+768*a^6*b^4*t^6*x^2-768*a^4*b^6*t^8-768*a^4*b^6*t^6*x^2+768*a^2*b^8*t^8+256*a^2*b^8*t^6*x^2-256*b^10*t^8+64*L^4*a^4*b^2*t^6-128*L^4*a^2*b^4*t^6+64*L^4*b^6*t^6+128*L^2*a^6*b^2*t^6+48*L^2*a^6*b^2*t^4*x^2-544*L^2*a^4*b^4*t^6-96*L^2*a^4*b^4*t^4*x^2+704*L^2*a^2*b^6*t^6+48*L^2*a^2*b^6*t^4*x^2-288*L^2*b^8*t^6-96*a^6*b^4*t^4*x^2+96*a^4*b^6*t^6+192*a^4*b^6*t^4*x^2-192*a^2*b^8*t^6-96*a^2*b^8*t^4*x^2+96*b^10*t^6-64*L^4*a^4*b^2*t^4+144*L^4*a^2*b^4*t^4-80*L^4*b^6*t^4+64*L^2*a^4*b^4*t^4+12*L^2*a^4*b^4*t^2*x^2-160*L^2*a^2*b^6*t^4-12*L^2*a^2*b^6*t^2*x^2+96*L^2*b^8*t^4-16*a^4*b^6*t^2*x^2+16*a^2*b^8*t^4+16*a^2*b^8*t^2*x^2-16*b^10*t^4+16*L^4*a^4*b^2*t^2-40*L^4*a^2*b^4*t^2+25*L^4*b^6*t^2+8*L^2*a^2*b^6*t^2+L^2*a^2*b^6*x^2-10*L^2*b^8*t^2-a^2*b^8*x^2+b^10*t^2＝0
$

然后令$a=5,b=3,L=4.615$

$150994944*t^10-419430400*t^8*x^2-1.160606122*10^8*t^8-9.63495526*10^7*t^6*x^2+1.146393024*10^8*t^6+9.11893248*10^6*t^4*x^2-3.548695956*10^7*t^4+3.615149880*10^6*t^2*x^2+
1.411660538*10^7*t^2+2.241351506*10^5*x^2=0$

画图得到


好像有问题啊！！

mathe

你现在只是得出x关于t的参数形式，还得得出y关于t的参数形式，然后就有了曲线的参数方程了。

wayne

以前好像算过类似于包络的问题吧?
http://zh.wikipedia.org/zh-cn/%E5%8C%85%E7%B5%A1%E7%B7%9A

wayne

直线方程是：
$\frac{\cosu}{\cosv}*\frac{x}{a}+\frac{\sinu}{\cosv}*\frac{y}{b} =1$
约束条件是：
$4*sinv^2*(a^2*sinu^2+b^2*cosu^2)=L^2$
消去$v$得 定长为L的线段簇为：
\(f(x,y,u) = 4(a^2 \sin^2u+b^2 \cos^2u)(a^2 b^2-(a y \sin u+b x \cos u)^2)-a^2 b^2 L^2\)

wayne

只能算出x,y关于u或者v的表达.
但u,v貌似很难同时消掉 (Mathematica一直在running).

\[ \left\{ \eqalign{ % by gxqcn
x &=\frac{a\cos u \left(3 a^4+3 b^4+2 a^2 b^2-2 a^2 L^2+ (-4 a^4+2 a^2 L^2+4 b^4-2 b^2 L^2)\cos (2 u)+(a^2-b^2)^2 \cos (4 u)\right)}{4\left (a^2 \sin ^2(u)+b^2 \cos ^2(u)\right)^2 \sqrt{4-\frac{L^2}{a^2 \sin ^2u+b^2 \cos ^2u}}} \\
y &=\frac{b \sin u\left (3 a^4+3 b^4+2 a^2 b^2-2 b^2 L^2+ (-4 a^4+2 a^2 L^2+4 b^4-2 b^2 L^2)\cos (2 u)+(a^2-b^2)^2 \cos (4 u)\right)}{4\left (a^2 \sin ^2(u)+b^2 \cos ^2(u)\right)^2 \sqrt{4-\frac{L^2}{a^2 \sin ^2u+b^2 \cos ^2u}}}
} \right.  \]

画图,得到椭圆定长弦的包络曲线图 (100条直线) 如下:

1. Manipulate[Show[{ContourPlot[Evaluate[Join[Table[4((a b)^2-(b x Cos[u]+a y Sin[u])^2)(a^2 Sin[u]^2+b^2 Cos[u]^2)-a^2 b^2 L^2==0,{u,0.,2Pi,1/n*2Pi}],{x^2/a^2+y^2/b^2==1}]],{x,-a,a},{y,-b,b},AspectRatio->1,PlotLabel->ToString@Row[{"a=",a,",b=",b,",L=",L}],PlotRange->{{-a,a},{-b,b}},ContourStyle->Evaluate[Flatten[{Table[Directive[Yellow,Thickness[0.0011]],{n+1}],Directive[Red,Thickness[0.01]]}]]],ParametricPlot[{(a Cos[u] (3 a^4+2 a^2 b^2+3 b^4-2 a^2 L^2+(-4 a^4+4 b^4+2 a^2 L^2-2 b^2 L^2) Cos[2 u]+(a^4-2 a^2 b^2+b^4) Cos[4 u]))/(4 Sqrt[4- (L^2)/(a^2 Sin[u]^2+b^2 Cos[u]^2)] (a^2 Sin[u]^2+b^2 Cos[u]^2)^2),(b Sin[u] (3 a^4+2 a^2 b^2+3 b^4-2 b^2 L^2+(-4 a^4+4 b^4+2 a^2 L^2-2 b^2 L^2) Cos[2 u]+(a^4-2 a^2 b^2+b^4) Cos[4 u]))/(4 Sqrt[4- (L^2)/(a^2 Sin[u]^2+b^2 Cos[u]^2)] (a^2 Sin[u]^2+b^2 Cos[u]^2)^2)},{u,0,2Pi},PlotRange->{{-a,a},{-b,b}},PlotStyle->Directive[Blue,Thickness[.01]]]}],{{a,5},1,20,1},{{b,3},1,a,1},{{L,a},1,2a,1},{{n,100},1,1000,10},ControlPlacement->Left]

复制代码

wayne

六年半之后的回复，哈哈哈，消元成功了
对于\(f(x,y,u) = 4(a^2 \sin^2u+b^2 \cos^2u)(a^2 b^2-(a y \sin u+b x \cos u)^2)-a^2 b^2 L^2\), 三角函数有理化$t=tanu$之后，
就是$4 a^2 b^4-a^2 b^2 L^2+4 a^4 b^2 t^2+4 a^2 b^4 t^2-2 a^2 b^2 L^2 t^2+4 a^4 b^2 t^4-a^2 b^2 L^2 t^4-4 b^4 x^2-4 a^2 b^2 t^2 x^2-8 a b^3 t x y-8 a^3 b t^3 x y-4 a^2 b^2 t^2 y^2-4 a^4 t^4 y^2=0$

1. 
   
2. 4((a b)^2-(b x Cos[u]+a y Sin[u])^2)(a^2 Sin[u]^2+b^2 Cos[u]^2)-a^2 b^2 L^2/.u->ArcTan[t]//TrigExpand//Factor
   
3. eq=4 a^2 b^4-a^2 b^2 L^2+4 a^4 b^2 t^2+4 a^2 b^4 t^2-2 a^2 b^2 L^2 t^2+4 a^4 b^2 t^4-a^2 b^2 L^2 t^4-4 b^4 x^2-4 a^2 b^2 t^2 x^2-8 a b^3 t x y-8 a^3 b t^3 x y-4 a^2 b^2 t^2 y^2-4 a^4 t^4 y^2;
   
4. GroebnerBasis[{eq==0,D[eq,t]==0},{},t,MonomialOrder->EliminationOrder]//Factor

复制代码

1. 16 a^16 b^8-64 a^14 b^10+96 a^12 b^12-64 a^10 b^14+16 a^8 b^16-4 a^16 b^6 L^2+12 a^14 b^8 L^2-8 a^12 b^10 L^2-8 a^10 b^12 L^2+12 a^8 b^14 L^2-4 a^6 b^16 L^2+a^14 b^6 L^4-4 a^12 b^8 L^4+6 a^10 b^10 L^4-4 a^8 b^12 L^4+a^6 b^14 L^4-80 a^14 b^8 x^2+256 a^12 b^10 x^2-288 a^10 b^12 x^2+128 a^8 b^14 x^2-16 a^6 b^16 x^2+16 a^14 b^6 L^2 x^2+4 a^12 b^8 L^2 x^2-112 a^10 b^10 L^2 x^2+152 a^8 b^12 L^2 x^2-64 a^6 b^14 L^2 x^2+4 a^4 b^16 L^2 x^2-12 a^12 b^6 L^4 x^2+28 a^10 b^8 L^4 x^2-12 a^8 b^10 L^4 x^2-12 a^6 b^12 L^4 x^2+8 a^4 b^14 L^4 x^2+2 a^10 b^6 L^6 x^2-6 a^8 b^8 L^6 x^2+6 a^6 b^10 L^6 x^2-2 a^4 b^12 L^6 x^2+160 a^12 b^8 x^4-384 a^10 b^10 x^4+288 a^8 b^12 x^4-64 a^6 b^14 x^4-24 a^12 b^6 L^2 x^4-88 a^10 b^8 L^2 x^4+296 a^8 b^10 L^2 x^4-232 a^6 b^12 L^2 x^4+48 a^4 b^14 L^2 x^4+22 a^10 b^6 L^4 x^4-4 a^8 b^8 L^4 x^4-66 a^6 b^10 L^4 x^4+56 a^4 b^12 L^4 x^4-8 a^2 b^14 L^4 x^4-8 a^8 b^6 L^6 x^4+12 a^6 b^8 L^6 x^4-4 a^2 b^12 L^6 x^4+a^6 b^6 L^8 x^4-2 a^4 b^8 L^8 x^4+a^2 b^10 L^8 x^4-160 a^10 b^8 x^6+256 a^8 b^10 x^6-96 a^6 b^12 x^6+16 a^10 b^6 L^2 x^6+120 a^8 b^8 L^2 x^6-224 a^6 b^10 L^2 x^6+88 a^4 b^12 L^2 x^6-12 a^8 b^6 L^4 x^6-28 a^6 b^8 L^4 x^6+72 a^4 b^10 L^4 x^6-32 a^2 b^12 L^4 x^6+2 a^6 b^6 L^6 x^6+2 a^4 b^8 L^6 x^6-8 a^2 b^10 L^6 x^6+4 b^12 L^6 x^6+80 a^8 b^8 x^8-64 a^6 b^10 x^8-4 a^8 b^6 L^2 x^8-52 a^6 b^8 L^2 x^8+48 a^4 b^10 L^2 x^8+a^6 b^6 L^4 x^8+8 a^4 b^8 L^4 x^8-8 a^2 b^10 L^4 x^8-16 a^6 b^8 x^10+4 a^4 b^8 L^2 x^10-16 a^16 b^6 y^2+128 a^14 b^8 y^2-288 a^12 b^10 y^2+256 a^10 b^12 y^2-80 a^8 b^14 y^2+4 a^16 b^4 L^2 y^2-64 a^14 b^6 L^2 y^2+152 a^12 b^8 L^2 y^2-112 a^10 b^10 L^2 y^2+4 a^8 b^12 L^2 y^2+16 a^6 b^14 L^2 y^2+8 a^14 b^4 L^4 y^2-12 a^12 b^6 L^4 y^2-12 a^10 b^8 L^4 y^2+28 a^8 b^10 L^4 y^2-12 a^6 b^12 L^4 y^2-2 a^12 b^4 L^6 y^2+6 a^10 b^6 L^6 y^2-6 a^8 b^8 L^6 y^2+2 a^6 b^10 L^6 y^2+64 a^14 b^6 x^2 y^2-320 a^12 b^8 x^2 y^2+512 a^10 b^10 x^2 y^2-320 a^8 b^12 x^2 y^2+64 a^6 b^14 x^2 y^2-12 a^14 b^4 L^2 x^2 y^2-12 a^12 b^6 L^2 x^2 y^2+24 a^10 b^8 L^2 x^2 y^2+24 a^8 b^10 L^2 x^2 y^2-12 a^6 b^12 L^2 x^2 y^2-12 a^4 b^14 L^2 x^2 y^2+22 a^12 b^4 L^4 x^2 y^2+4 a^10 b^6 L^4 x^2 y^2-52 a^8 b^8 L^4 x^2 y^2+4 a^6 b^10 L^4 x^2 y^2+22 a^4 b^12 L^4 x^2 y^2-12 a^10 b^4 L^6 x^2 y^2+12 a^8 b^6 L^6 x^2 y^2+12 a^6 b^8 L^6 x^2 y^2-12 a^4 b^10 L^6 x^2 y^2+2 a^8 b^4 L^8 x^2 y^2-4 a^6 b^6 L^8 x^2 y^2+2 a^4 b^8 L^8 x^2 y^2-96 a^12 b^6 x^4 y^2+192 a^10 b^8 x^4 y^2-160 a^8 b^10 x^4 y^2+64 a^6 b^12 x^4 y^2+12 a^12 b^4 L^2 x^4 y^2+224 a^10 b^6 L^2 x^4 y^2-368 a^8 b^8 L^2 x^4 y^2+80 a^6 b^10 L^2 x^4 y^2+52 a^4 b^12 L^2 x^4 y^2-32 a^10 b^4 L^4 x^4 y^2-40 a^8 b^6 L^4 x^4 y^2+136 a^6 b^8 L^4 x^4 y^2-44 a^4 b^10 L^4 x^4 y^2-20 a^2 b^12 L^4 x^4 y^2+8 a^8 b^4 L^6 x^4 y^2-4 a^6 b^6 L^6 x^4 y^2-14 a^4 b^8 L^6 x^4 y^2+10 a^2 b^10 L^6 x^4 y^2+64 a^10 b^6 x^6 y^2+64 a^8 b^8 x^6 y^2-64 a^6 b^10 x^6 y^2-4 a^10 b^4 L^2 x^6 y^2-156 a^8 b^6 L^2 x^6 y^2+180 a^6 b^8 L^2 x^6 y^2-52 a^4 b^10 L^2 x^6 y^2+2 a^8 b^4 L^4 x^6 y^2+36 a^6 b^6 L^4 x^6 y^2-54 a^4 b^8 L^4 x^6 y^2+20 a^2 b^10 L^4 x^6 y^2-16 a^8 b^6 x^8 y^2-64 a^6 b^8 x^8 y^2+8 a^6 b^6 L^2 x^8 y^2+12 a^4 b^8 L^2 x^8 y^2-64 a^14 b^6 y^4+288 a^12 b^8 y^4-384 a^10 b^10 y^4+160 a^8 b^12 y^4+48 a^14 b^4 L^2 y^4-232 a^12 b^6 L^2 y^4+296 a^10 b^8 L^2 y^4-88 a^8 b^10 L^2 y^4-24 a^6 b^12 L^2 y^4-8 a^14 b^2 L^4 y^4+56 a^12 b^4 L^4 y^4-66 a^10 b^6 L^4 y^4-4 a^8 b^8 L^4 y^4+22 a^6 b^10 L^4 y^4-4 a^12 b^2 L^6 y^4+12 a^8 b^6 L^6 y^4-8 a^6 b^8 L^6 y^4+a^10 b^2 L^8 y^4-2 a^8 b^4 L^8 y^4+a^6 b^6 L^8 y^4+64 a^12 b^6 x^2 y^4-160 a^10 b^8 x^2 y^4+192 a^8 b^10 x^2 y^4-96 a^6 b^12 x^2 y^4+52 a^12 b^4 L^2 x^2 y^4+80 a^10 b^6 L^2 x^2 y^4-368 a^8 b^8 L^2 x^2 y^4+224 a^6 b^10 L^2 x^2 y^4+12 a^4 b^12 L^2 x^2 y^4-20 a^12 b^2 L^4 x^2 y^4-44 a^10 b^4 L^4 x^2 y^4+136 a^8 b^6 L^4 x^2 y^4-40 a^6 b^8 L^4 x^2 y^4-32 a^4 b^10 L^4 x^2 y^4+10 a^10 b^2 L^6 x^2 y^4-14 a^8 b^4 L^6 x^2 y^4-4 a^6 b^6 L^6 x^2 y^4+8 a^4 b^8 L^6 x^2 y^4+64 a^10 b^6 x^4 y^4-32 a^8 b^8 x^4 y^4+64 a^6 b^10 x^4 y^4-104 a^10 b^4 L^2 x^4 y^4+80 a^8 b^6 L^2 x^4 y^4+80 a^6 b^8 L^2 x^4 y^4-104 a^4 b^10 L^2 x^4 y^4+a^10 b^2 L^4 x^4 y^4+48 a^8 b^4 L^4 x^4 y^4-92 a^6 b^6 L^4 x^4 y^4+48 a^4 b^8 L^4 x^4 y^4+a^2 b^10 L^4 x^4 y^4-64 a^8 b^6 x^6 y^4-96 a^6 b^8 x^6 y^4+4 a^8 b^4 L^2 x^6 y^4+24 a^6 b^6 L^2 x^6 y^4+12 a^4 b^8 L^2 x^6 y^4-96 a^12 b^6 y^6+256 a^10 b^8 y^6-160 a^8 b^10 y^6+88 a^12 b^4 L^2 y^6-224 a^10 b^6 L^2 y^6+120 a^8 b^8 L^2 y^6+16 a^6 b^10 L^2 y^6-32 a^12 b^2 L^4 y^6+72 a^10 b^4 L^4 y^6-28 a^8 b^6 L^4 y^6-12 a^6 b^8 L^4 y^6+4 a^12 L^6 y^6-8 a^10 b^2 L^6 y^6+2 a^8 b^4 L^6 y^6+2 a^6 b^6 L^6 y^6-64 a^10 b^6 x^2 y^6+64 a^8 b^8 x^2 y^6+64 a^6 b^10 x^2 y^6-52 a^10 b^4 L^2 x^2 y^6+180 a^8 b^6 L^2 x^2 y^6-156 a^6 b^8 L^2 x^2 y^6-4 a^4 b^10 L^2 x^2 y^6+20 a^10 b^2 L^4 x^2 y^6-54 a^8 b^4 L^4 x^2 y^6+36 a^6 b^6 L^4 x^2 y^6+2 a^4 b^8 L^4 x^2 y^6-96 a^8 b^6 x^4 y^6-64 a^6 b^8 x^4 y^6+12 a^8 b^4 L^2 x^4 y^6+24 a^6 b^6 L^2 x^4 y^6+4 a^4 b^8 L^2 x^4 y^6-64 a^10 b^6 y^8+80 a^8 b^8 y^8+48 a^10 b^4 L^2 y^8-52 a^8 b^6 L^2 y^8-4 a^6 b^8 L^2 y^8-8 a^10 b^2 L^4 y^8+8 a^8 b^4 L^4 y^8+a^6 b^6 L^4 y^8-64 a^8 b^6 x^2 y^8-16 a^6 b^8 x^2 y^8+12 a^8 b^4 L^2 x^2 y^8+8 a^6 b^6 L^2 x^2 y^8-16 a^8 b^6 y^10+4 a^8 b^4 L^2 y^10=0

复制代码

跟数学星空在https://bbs.emath.ac.cn/thread-4289-2-1.html  19#的计算结果一致。