极值条件:1375*x^6-68050*x^5+1307185*x^4-12098490*x^3+52655856*x^2-75334734*x-44362866=0
解得实根 x_1=4.36005201173273 , alpha+beta=155.3292570
x_2=12.9496308788891,alpha+beta=63.18348125
x_3=14.8273178342319,alpha+beta=43.02405272
极值条件:
(322+40*x^3-60*x^2-624*x)/((2*x-1)*(11240+25*x^4-510*x^3+3677*x^2-10828*x)*arccos((106-51*x+5*x^2)/sqrt(11240+25*x^4-510*x^3+3677*x^2-10828*x))^2)+
(-1079875+405*x^3-16875*x^2+234001*x)/((9*x-125)*(878666+25*x^4-1360*x^3+27867*x^2-254938*x)*arccos((929-136*x+5*x^2)/sqrt(878666+25*x^4-1360*x^3+27867*x^2-254938*x))^2)=0
这就是为什么振荡厉害的原因! 取四个点(A,B,C,D)
x_1 = 2, x_2 = 8, x_3 = 5, x_4 = 40, y_1 = 9, y_2 = 30, y_3 = 13, y_4 = 70可以得到:
取极值条件:125*x^6+825*x^5-20545*x^4-4370985*x^3+80880055*x^2-481882662*x+959634841=0
解得实根: x_1=8.48768754238868,alpha+beta=317.9909271
x_2=24.3626672156265,alpha+beta=181.0623841
现已找到在直线L(y=k*x+a)同侧有依次为A(x1,y1),B(x2,y2),C(x3,y3),D(x4,y4)四个点,AB,CD对L上一点的张角和(alpha+beta)有如下结论:
设f(x1,x2)=((-2*y2*k^2-2*x1*k-2*x1*k^3+2*x2*k^3+2*x2*k-2*y2+2*y1+2*y1*k^2)*x^2+(-4*k^2*y1*x2-4*a*x1+4*x2*a*k^2-4*x1*k^2*a-
4*y1*x2+4*x1*y2+4*a*x2+4*k^2*x1*y2)*x-4*y1*a*k*x2+2*y1*x2^2-2*a*y2^2+2*a^2*y2+2*y1^2*a-2*y1*a^2+2*x1^2*a-2*a*x2^2-
2*x1*k*x2^2-2*x1^2*y2-2*x1*a^2*k+2*y1*y2^2+2*x2*a^2*k-2*x1*k*y2^2+2*k*x1^2*x2+2*k*y1^2*x2+4*x1*k*a*y2-2*y1^2*y2)
h(x1,x2)=(k^4+2*k^2+1)*x^4+(-2*x1*k^2-2*x1-2*y1*k^3-2*y1*k+4*k^3*a+4*k*a-2*k^2*x2-2*k^3*y2-2*x2-2*y2*k)*x^3+
(-4*x1*k*a+4*x1*y2*k-6*y1*a*k^2+4*y1*k*x2-4*k*a*x2+4*y1*k^2*y2-6*k^2*a*y2+x1^2+x2^2-2*y1*a+2*a^2-
2*y2*a+4*x1*x2+k^2*x2^2+k^2*y2^2+x1^2*k^2+y1^2*k^2+6*a^2*k^2+y2^2+y1^2)*x^2+(-2*x1^2*x2-2*x1^2*y2*k+
2*x1^2*k*a-2*y1^2*x2-2*y1^2*y2*k+2*y1^2*k*a+4*y1*a*x2+8*y1*a*y2*k-6*y1*a^2*k-2*a^2*x2-6*a^2*y2*k+4*a^3*k-
2*x1*x2^2-2*x1*y2^2+4*x1*y2*a-2*x1*a^2-2*y1*k*x2^2-2*y1*k*y2^2+2*k*a*x2^2+2*k*a*y2^2)*x+x1^2*x2^2+x1^2*y2^2-
2*x1^2*y2*a+x1^2*a^2+y1^2*x2^2+y1^2*y2^2-2*y1^2*y2*a+y1^2*a^2-2*y1*a*x2^2-2*y1*a*y2^2+4*y1*a^2*y2-
2*y1*a^3+a^2*x2^2+a^2*y2^2-2*a^3*y2+a^4
g(x1,x2)=(((y2-y1)-k*(x2-x1))*x+a*(x1-x2)+(x2*y1-x1*y2))/|(((y2-y1)-k*(x2-x1))*x+a*(x1-x2)+(x2*y1-x1*y2))|
则alpha+beta取极值条件为
{f(x1,x2)*g(x1,x2)}/{h(x1,x2)}+{f(x3,x4)*g(x3,x4)}/{h(x3,x4)}=0 ,且x为方程的实根.
alpha+beta=(180/pi)*(arccos(((x1-x)^2+(y1-k*x-a)^2+(x2-x)^2+(y2-k*x-a)^2-(x1-x2)^2-(y1-y2)^2)/(2*sqrt(((x1-x)^2+(y1-y)^2)*((x2-x)^2+(y2-y)^2))))+
(180/pi)*(arccos(((x1-x)^2+(y1-k*x-a)^2+(x2-x)^2+(y2-k*x-a)^2-(x1-x2)^2-(y1-y2)^2)/(2*sqrt(((x1-x)^2+(y1-k*x-a)^2)*((x2-x)^2+(y2-k*x-a)^2))))) 对于(2)我们也很容易得到1/alpha+1/beta存在极值的条件为:
{f(x1,x2)*g(x1,x2)}/{(arccos({p(x1,x2)}/{sqrt(h(x1,x2))}))^2*h(x1,x2)}+{f(x3,x4)*g(x3,x4)}/{(arccos({p(x3,x4)}/{sqrt(h(x3,x4))}))^2*h(x3,x4)}=0
且p(x1,x2)=(k^2+1)*x^2+(-x1-y2*k+2*k*a-y1*k-x2)*x-y2*a+a^2-y1*a+x1*x2+y1*y2 若ABCD的连线平行于L,且L为水平线时:
k=0,y1=y2=y3=y4=d
代入化简各式有:
f(x1,x2)=-(2*(x1-x2))*(2*x-x1-x2)*(a-d)
h(x1,x2)=(x2^2+d^2-2*d*a+a^2)*(x1^2+d^2-2*d*a+a^2)-2*(x2+x1)*(a^2+x1*x2+d^2-2*d*a)*x+
(x1^2+2*d^2-4*d*a+2*a^2+4*x1*x2+x2^2)*x^2+(-2*x1-2*x2)*x^3+x^4
g(x1,x2)={(x1-x2)*(a-d)}/{|(x1-x2)*(a-d)|}
p(x1,x2)=x^2+(-x1-x2)*x-2*d*a+a^2+x1*x2+d^2 上面几贴，是否说明已得到最终的公式解了？确认一下。
比较复杂，估计难以通过尺规作图确定出该点吧。 太复杂了，我只有看的份，呵呵
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