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小木虫论坛-学术科研互动平台 » 计算模拟区 » 程序语言 » 其它 » matlab求解非线性方程组，求助！
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Kevin1987s

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[求助] matlab求解非线性方程组，求助！

θ1-4未知，用已知表示未知，怎么编程呢？
这是我编的程序，方程中用l代替l1,m代替l2
>> syms \alpha x y z R r l m \theta_1 \theta_2 \theta_3 \theta_4
eq1=[2^1/2*r*sin(\pi/4-\alpha)*cos\alpha+x-R-l*sin\theta_1]^2+[2^1/2*r*sin(\pi/4-\alpha)*sin\alpha+y]^2+(z+l*cos\theta_1)^2-m^2;
eq2=[-2^1/2*r*cos(\pi/4-\alpha)*sin\alpha+x]^2+[2^1/2*r*cos(\pi/4-\alpha)*cos\alpha+y-R-l*sin\theta_2]^2+(z+l*cos\theta_2)^2-m^2;
eq3=[-2^1/2*r*sin(\pi/4-\alpha)*cos\alpha+x+R+l*sin\theta_3]^2+[-2^1/2*r*sin(\pi/4-\alpha)*sin\alpha+y]^2+(z+l*cos\theta_3)^2-m^2;
eq4=[2^1/2*r*cos(\pi/4-\alpha)*sin\alpha+x]^2+[-2^1/2*r*cos(\pi/4-\alpha)*cos\alpha+y+R+l*sin\theta_4]^2+(z+l*cos\theta_4)^2-m^2;
d=solve(eq1,eq2,eq3,eq4)
d.\theta_1 \theta_2 \theta_3 \theta_4
??? Error using ==> syms at 61
Not a valid variable name.
应该怎么改啊？十分感谢！

方程组
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1楼 2012-05-21 12:21:08
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csgt0

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【答案】应助回帖

感谢参与，应助指数 +1
变量名不对吧，别用\应该可以
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showmethemoney
2楼2012-05-21 13:29:15
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Kevin1987s

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引用回帖:
2楼: Originally posted by csgt0 at 2012-05-21 13:29:15:
变量名不对吧，别用\应该可以

可能是方程太复杂了，一直Busy。。。。有优化的算法么？谢谢您！
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3楼2012-05-21 14:15:38
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csgt0

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我也不太懂，刚用你的2,4式试了下，能解出x，y，但是也是无穷串
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showmethemoney
4楼2012-05-21 14:20:32
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Kevin1987s

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引用回帖:
4楼: Originally posted by csgt0 at 2012-05-21 14:20:32:
我也不太懂，刚用你的2,4式试了下，能解出x，y，但是也是无穷串

限制有效数字呢？在倒数第二行加上vpa(theta_1,4),vpa(theta_2,4),vpa(theta_3,4),vpa(theta_4,4)
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5楼2012-05-21 14:28:47
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csgt0

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你是要解theta1234啊，那一个一个解就可以得出来，相当于你这4个方程是独立的。vpa只对具体数字管用，对符号不管用吧
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showmethemoney
6楼2012-05-21 14:38:22
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csgt0

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d=solve(eq1,'theta_1')
d =

  2*atan((2*(2*l*x - 2*R*l + ((4*l*x - 4*R*l + 2*2^(1/2)*l*r - 8*R*l*tan(alpha/2)^2 - 4*R*l*tan(alpha/2)^4 + 8*l*x*tan(alpha/2)^2 + 4*l*x*tan(alpha/2)^4 - 4*2^(1/2)*l*r*tan(alpha/2)^2 + 4*2^(1/2)*l*r*tan(alpha/2)^3 + 2*2^(1/2)*l*r*tan(alpha/2)^4 - 4*2^(1/2)*l*r*tan(alpha/2))^2/4 - ((r^2*cos(alpha)^4)/2 - 2*l*z - 2*R*x + 2*R^2*tan(alpha/2)^2 + R^2*tan(alpha/2)^4 + R^2 + 2*l^2*tan(alpha/2)^2 + l^2*tan(alpha/2)^4 - 2*m^2*tan(alpha/2)^2 - m^2*tan(alpha/2)^4 + l^2 + 2*x^2*tan(alpha/2)^2 + x^2*tan(alpha/2)^4 - m^2 + 2*y^2*tan(alpha/2)^2 + y^2*tan(alpha/2)^4 + 2*z^2*tan(alpha/2)^2 + z^2*tan(alpha/2)^4 + x^2 + y^2 + z^2 - r^2*cos(alpha)^3*sin(alpha) + (r^2*cos(alpha)^2*sin(alpha)^2)/2 + r^2*sin(alpha)^2*sin(alpha - pi/4)^2 - 4*R*x*tan(alpha/2)^2 - 2*R*x*tan(alpha/2)^4 - 4*l*z*tan(alpha/2)^2 - 2*l*z*tan(alpha/2)^4 + r^2*tan(alpha/2)^2*cos(alpha)^4 + (r^2*tan(alpha/2)^4*cos(alpha)^4)/2 - 2*r*y*sin(alpha)*sin(alpha - pi/4) + r^2*tan(alpha/2)^2*cos(alpha)^2*sin(alpha)^2 + (r^2*tan(alpha/2)^4*cos(alpha)^2*sin(alpha)^2)/2 + 2*r^2*tan(alpha/2)^2*sin(alpha)^2*sin(alpha - pi/4)^2 + r^2*tan(alpha/2)^4*sin(alpha)^2*sin(alpha - pi/4)^2 - 2^(1/2)*R*r*cos(alpha)^2 - 2*r^2*tan(alpha/2)^2*cos(alpha)^3*sin(alpha) - r^2*tan(alpha/2)^4*cos(alpha)^3*sin(alpha) + 2^(1/2)*r*x*cos(alpha)^2 + 2^(1/2)*R*r*cos(alpha)*sin(alpha) - 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)^2 - 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)^2 - 2^(1/2)*r*x*cos(alpha)*sin(alpha) + 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)^2 + 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)^2 - 4*r*y*tan(alpha/2)^2*sin(alpha)*sin(alpha - pi/4) - 2*r*y*tan(alpha/2)^4*sin(alpha)*sin(alpha - pi/4) + 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)*sin(alpha) + 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)*sin(alpha) - 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)*sin(alpha) - 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)*sin(alpha))*(2*l*z - 2*R*x + (r^2*cos(alpha)^4)/2 + 2*R^2*tan(alpha/2)^2 + R^2*tan(alpha/2)^4 + R^2 + 2*l^2*tan(alpha/2)^2 + l^2*tan(alpha/2)^4 - 2*m^2*tan(alpha/2)^2 - m^2*tan(alpha/2)^4 + l^2 + 2*x^2*tan(alpha/2)^2 + x^2*tan(alpha/2)^4 - m^2 + 2*y^2*tan(alpha/2)^2 + y^2*tan(alpha/2)^4 + 2*z^2*tan(alpha/2)^2 + z^2*tan(alpha/2)^4 + x^2 + y^2 + z^2 - r^2*cos(alpha)^3*sin(alpha) + (r^2*cos(alpha)^2*sin(alpha)^2)/2 + r^2*sin(alpha)^2*sin(alpha - pi/4)^2 - 4*R*x*tan(alpha/2)^2 - 2*R*x*tan(alpha/2)^4 + 4*l*z*tan(alpha/2)^2 + 2*l*z*tan(alpha/2)^4 + r^2*tan(alpha/2)^2*cos(alpha)^4 + (r^2*tan(alpha/2)^4*cos(alpha)^4)/2 - 2*r*y*sin(alpha)*sin(alpha - pi/4) + r^2*tan(alpha/2)^2*cos(alpha)^2*sin(alpha)^2 + (r^2*tan(alpha/2)^4*cos(alpha)^2*sin(alpha)^2)/2 + 2*r^2*tan(alpha/2)^2*sin(alpha)^2*sin(alpha - pi/4)^2 + r^2*tan(alpha/2)^4*sin(alpha)^2*sin(alpha - pi/4)^2 - 2^(1/2)*R*r*cos(alpha)^2 - 2*r^2*tan(alpha/2)^2*cos(alpha)^3*sin(alpha) - r^2*tan(alpha/2)^4*cos(alpha)^3*sin(alpha) + 2^(1/2)*r*x*cos(alpha)^2 + 2^(1/2)*R*r*cos(alpha)*sin(alpha) - 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)^2 - 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)^2 - 2^(1/2)*r*x*cos(alpha)*sin(alpha) + 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)^2 + 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)^2 - 4*r*y*tan(alpha/2)^2*sin(alpha)*sin(alpha - pi/4) - 2*r*y*tan(alpha/2)^4*sin(alpha)*sin(alpha - pi/4) + 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)*sin(alpha) + 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)*sin(alpha) - 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)*sin(alpha) - 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)*sin(alpha)))^(1/2) + 2^(1/2)*l*r - 4*R*l*tan(alpha/2)^2 - 2*R*l*tan(alpha/2)^4 + 4*l*x*tan(alpha/2)^2 + 2*l*x*tan(alpha/2)^4 - 2*2^(1/2)*l*r*tan(alpha/2)^2 + 2*2^(1/2)*l*r*tan(alpha/2)^3 + 2^(1/2)*l*r*tan(alpha/2)^4 - 2*2^(1/2)*l*r*tan(alpha/2)))/(r^2*cos(alpha)^4 - 4*l*z - 4*R*x + 4*R^2*tan(alpha/2)^2 + 2*R^2*tan(alpha/2)^4 + 2*R^2 + 4*l^2*tan(alpha/2)^2 + 2*l^2*tan(alpha/2)^4 - 4*m^2*tan(alpha/2)^2 - 2*m^2*tan(alpha/2)^4 + 2*l^2 + 4*x^2*tan(alpha/2)^2 + 2*x^2*tan(alpha/2)^4 - 2*m^2 + 4*y^2*tan(alpha/2)^2 + 2*y^2*tan(alpha/2)^4 + 4*z^2*tan(alpha/2)^2 + 2*z^2*tan(alpha/2)^4 + 2*x^2 + 2*y^2 + 2*z^2 - 2*r^2*cos(alpha)^3*sin(alpha) + r^2*cos(alpha)^2*sin(alpha)^2 + 2*r^2*sin(alpha)^2*sin(alpha - pi/4)^2 - 8*R*x*tan(alpha/2)^2 - 4*R*x*tan(alpha/2)^4 - 8*l*z*tan(alpha/2)^2 - 4*l*z*tan(alpha/2)^4 + 2*r^2*tan(alpha/2)^2*cos(alpha)^4 + r^2*tan(alpha/2)^4*cos(alpha)^4 - 4*r*y*sin(alpha)*sin(alpha - pi/4) + 2*r^2*tan(alpha/2)^2*cos(alpha)^2*sin(alpha)^2 + r^2*tan(alpha/2)^4*cos(alpha)^2*sin(alpha)^2 + 4*r^2*tan(alpha/2)^2*sin(alpha)^2*sin(alpha - pi/4)^2 + 2*r^2*tan(alpha/2)^4*sin(alpha)^2*sin(alpha - pi/4)^2 - 2*2^(1/2)*R*r*cos(alpha)^2 - 4*r^2*tan(alpha/2)^2*cos(alpha)^3*sin(alpha) - 2*r^2*tan(alpha/2)^4*cos(alpha)^3*sin(alpha) + 2*2^(1/2)*r*x*cos(alpha)^2 + 2*2^(1/2)*R*r*cos(alpha)*sin(alpha) - 4*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)^2 - 2*2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)^2 - 2*2^(1/2)*r*x*cos(alpha)*sin(alpha) + 4*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)^2 + 2*2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)^2 - 8*r*y*tan(alpha/2)^2*sin(alpha)*sin(alpha - pi/4) - 4*r*y*tan(alpha/2)^4*sin(alpha)*sin(alpha - pi/4) + 4*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)*sin(alpha) + 2*2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)*sin(alpha) - 4*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)*sin(alpha) - 2*2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)*sin(alpha)))
-2*atan((2*(2*R*l - 2*l*x + ((4*l*x - 4*R*l + 2*2^(1/2)*l*r - 8*R*l*tan(alpha/2)^2 - 4*R*l*tan(alpha/2)^4 + 8*l*x*tan(alpha/2)^2 + 4*l*x*tan(alpha/2)^4 - 4*2^(1/2)*l*r*tan(alpha/2)^2 + 4*2^(1/2)*l*r*tan(alpha/2)^3 + 2*2^(1/2)*l*r*tan(alpha/2)^4 - 4*2^(1/2)*l*r*tan(alpha/2))^2/4 - ((r^2*cos(alpha)^4)/2 - 2*l*z - 2*R*x + 2*R^2*tan(alpha/2)^2 + R^2*tan(alpha/2)^4 + R^2 + 2*l^2*tan(alpha/2)^2 + l^2*tan(alpha/2)^4 - 2*m^2*tan(alpha/2)^2 - m^2*tan(alpha/2)^4 + l^2 + 2*x^2*tan(alpha/2)^2 + x^2*tan(alpha/2)^4 - m^2 + 2*y^2*tan(alpha/2)^2 + y^2*tan(alpha/2)^4 + 2*z^2*tan(alpha/2)^2 + z^2*tan(alpha/2)^4 + x^2 + y^2 + z^2 - r^2*cos(alpha)^3*sin(alpha) + (r^2*cos(alpha)^2*sin(alpha)^2)/2 + r^2*sin(alpha)^2*sin(alpha - pi/4)^2 - 4*R*x*tan(alpha/2)^2 - 2*R*x*tan(alpha/2)^4 - 4*l*z*tan(alpha/2)^2 - 2*l*z*tan(alpha/2)^4 + r^2*tan(alpha/2)^2*cos(alpha)^4 + (r^2*tan(alpha/2)^4*cos(alpha)^4)/2 - 2*r*y*sin(alpha)*sin(alpha - pi/4) + r^2*tan(alpha/2)^2*cos(alpha)^2*sin(alpha)^2 + (r^2*tan(alpha/2)^4*cos(alpha)^2*sin(alpha)^2)/2 + 2*r^2*tan(alpha/2)^2*sin(alpha)^2*sin(alpha - pi/4)^2 + r^2*tan(alpha/2)^4*sin(alpha)^2*sin(alpha - pi/4)^2 - 2^(1/2)*R*r*cos(alpha)^2 - 2*r^2*tan(alpha/2)^2*cos(alpha)^3*sin(alpha) - r^2*tan(alpha/2)^4*cos(alpha)^3*sin(alpha) + 2^(1/2)*r*x*cos(alpha)^2 + 2^(1/2)*R*r*cos(alpha)*sin(alpha) - 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)^2 - 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)^2 - 2^(1/2)*r*x*cos(alpha)*sin(alpha) + 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)^2 + 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)^2 - 4*r*y*tan(alpha/2)^2*sin(alpha)*sin(alpha - pi/4) - 2*r*y*tan(alpha/2)^4*sin(alpha)*sin(alpha - pi/4) + 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)*sin(alpha) + 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)*sin(alpha) - 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)*sin(alpha) - 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)*sin(alpha))*(2*l*z - 2*R*x + (r^2*cos(alpha)^4)/2 + 2*R^2*tan(alpha/2)^2 + R^2*tan(alpha/2)^4 + R^2 + 2*l^2*tan(alpha/2)^2 + l^2*tan(alpha/2)^4 - 2*m^2*tan(alpha/2)^2 - m^2*tan(alpha/2)^4 + l^2 + 2*x^2*tan(alpha/2)^2 + x^2*tan(alpha/2)^4 - m^2 + 2*y^2*tan(alpha/2)^2 + y^2*tan(alpha/2)^4 + 2*z^2*tan(alpha/2)^2 + z^2*tan(alpha/2)^4 + x^2 + y^2 + z^2 - r^2*cos(alpha)^3*sin(alpha) + (r^2*cos(alpha)^2*sin(alpha)^2)/2 + r^2*sin(alpha)^2*sin(alpha - pi/4)^2 - 4*R*x*tan(alpha/2)^2 - 2*R*x*tan(alpha/2)^4 + 4*l*z*tan(alpha/2)^2 + 2*l*z*tan(alpha/2)^4 + r^2*tan(alpha/2)^2*cos(alpha)^4 + (r^2*tan(alpha/2)^4*cos(alpha)^4)/2 - 2*r*y*sin(alpha)*sin(alpha - pi/4) + r^2*tan(alpha/2)^2*cos(alpha)^2*sin(alpha)^2 + (r^2*tan(alpha/2)^4*cos(alpha)^2*sin(alpha)^2)/2 + 2*r^2*tan(alpha/2)^2*sin(alpha)^2*sin(alpha - pi/4)^2 + r^2*tan(alpha/2)^4*sin(alpha)^2*sin(alpha - pi/4)^2 - 2^(1/2)*R*r*cos(alpha)^2 - 2*r^2*tan(alpha/2)^2*cos(alpha)^3*sin(alpha) - r^2*tan(alpha/2)^4*cos(alpha)^3*sin(alpha) + 2^(1/2)*r*x*cos(alpha)^2 + 2^(1/2)*R*r*cos(alpha)*sin(alpha) - 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)^2 - 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)^2 - 2^(1/2)*r*x*cos(alpha)*sin(alpha) + 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)^2 + 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)^2 - 4*r*y*tan(alpha/2)^2*sin(alpha)*sin(alpha - pi/4) - 2*r*y*tan(alpha/2)^4*sin(alpha)*sin(alpha - pi/4) + 2*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)*sin(alpha) + 2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)*sin(alpha) - 2*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)*sin(alpha) - 2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)*sin(alpha)))^(1/2) - 2^(1/2)*l*r + 4*R*l*tan(alpha/2)^2 + 2*R*l*tan(alpha/2)^4 - 4*l*x*tan(alpha/2)^2 - 2*l*x*tan(alpha/2)^4 + 2*2^(1/2)*l*r*tan(alpha/2)^2 - 2*2^(1/2)*l*r*tan(alpha/2)^3 - 2^(1/2)*l*r*tan(alpha/2)^4 + 2*2^(1/2)*l*r*tan(alpha/2)))/(r^2*cos(alpha)^4 - 4*l*z - 4*R*x + 4*R^2*tan(alpha/2)^2 + 2*R^2*tan(alpha/2)^4 + 2*R^2 + 4*l^2*tan(alpha/2)^2 + 2*l^2*tan(alpha/2)^4 - 4*m^2*tan(alpha/2)^2 - 2*m^2*tan(alpha/2)^4 + 2*l^2 + 4*x^2*tan(alpha/2)^2 + 2*x^2*tan(alpha/2)^4 - 2*m^2 + 4*y^2*tan(alpha/2)^2 + 2*y^2*tan(alpha/2)^4 + 4*z^2*tan(alpha/2)^2 + 2*z^2*tan(alpha/2)^4 + 2*x^2 + 2*y^2 + 2*z^2 - 2*r^2*cos(alpha)^3*sin(alpha) + r^2*cos(alpha)^2*sin(alpha)^2 + 2*r^2*sin(alpha)^2*sin(alpha - pi/4)^2 - 8*R*x*tan(alpha/2)^2 - 4*R*x*tan(alpha/2)^4 - 8*l*z*tan(alpha/2)^2 - 4*l*z*tan(alpha/2)^4 + 2*r^2*tan(alpha/2)^2*cos(alpha)^4 + r^2*tan(alpha/2)^4*cos(alpha)^4 - 4*r*y*sin(alpha)*sin(alpha - pi/4) + 2*r^2*tan(alpha/2)^2*cos(alpha)^2*sin(alpha)^2 + r^2*tan(alpha/2)^4*cos(alpha)^2*sin(alpha)^2 + 4*r^2*tan(alpha/2)^2*sin(alpha)^2*sin(alpha - pi/4)^2 + 2*r^2*tan(alpha/2)^4*sin(alpha)^2*sin(alpha - pi/4)^2 - 2*2^(1/2)*R*r*cos(alpha)^2 - 4*r^2*tan(alpha/2)^2*cos(alpha)^3*sin(alpha) - 2*r^2*tan(alpha/2)^4*cos(alpha)^3*sin(alpha) + 2*2^(1/2)*r*x*cos(alpha)^2 + 2*2^(1/2)*R*r*cos(alpha)*sin(alpha) - 4*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)^2 - 2*2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)^2 - 2*2^(1/2)*r*x*cos(alpha)*sin(alpha) + 4*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)^2 + 2*2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)^2 - 8*r*y*tan(alpha/2)^2*sin(alpha)*sin(alpha - pi/4) - 4*r*y*tan(alpha/2)^4*sin(alpha)*sin(alpha - pi/4) + 4*2^(1/2)*R*r*tan(alpha/2)^2*cos(alpha)*sin(alpha) + 2*2^(1/2)*R*r*tan(alpha/2)^4*cos(alpha)*sin(alpha) - 4*2^(1/2)*r*x*tan(alpha/2)^2*cos(alpha)*sin(alpha) - 2*2^(1/2)*r*x*tan(alpha/2)^4*cos(alpha)*sin(alpha)))

simple(d)

simplify:

  2*atan((2*cos(alpha/2)^4*(2*l*x - 2*R*l + ((16*l^2*(2*R - 2*x - 2^(1/2)*r*cos(alpha)^2 + (2^(1/2)*r*sin(2*alpha))/2)^2)/(cos(alpha) + 1)^4 - ((2*R^2 - 4*l*z - r^2*sin(2*alpha) - 4*R*x + 2*l^2 - 2*m^2 + r^2 + 2*x^2 + 2*y^2 + 2*z^2 - 2^(1/2)*R*r + 2*r*y*sin(pi/4 + 2*alpha) + 2^(1/2)*r*x - 2^(1/2)*r*y - 2^(1/2)*R*r*cos(2*alpha) + 2^(1/2)*R*r*sin(2*alpha) + 2^(1/2)*r*x*cos(2*alpha) - 2^(1/2)*r*x*sin(2*alpha))*(4*l*z - 4*R*x - r^2*sin(2*alpha) + 2*R^2 + 2*l^2 - 2*m^2 + r^2 + 2*x^2 + 2*y^2 + 2*z^2 - 2^(1/2)*R*r + 2*r*y*sin(pi/4 + 2*alpha) + 2^(1/2)*r*x - 2^(1/2)*r*y - 2^(1/2)*R*r*cos(2*alpha) + 2^(1/2)*R*r*sin(2*alpha) + 2^(1/2)*r*x*cos(2*alpha) - 2^(1/2)*r*x*sin(2*alpha)))/(4*cos(alpha/2)^8))^(1/2) + 2^(1/2)*l*r - 4*R*l*tan(alpha/2)^2 - 2*R*l*tan(alpha/2)^4 + 4*l*x*tan(alpha/2)^2 + 2*l*x*tan(alpha/2)^4 - 2*2^(1/2)*l*r*tan(alpha/2)^2 + 2*2^(1/2)*l*r*tan(alpha/2)^3 + 2^(1/2)*l*r*tan(alpha/2)^4 - 2*2^(1/2)*l*r*tan(alpha/2)))/(2*R^2 - 4*l*z - r^2*sin(2*alpha) - 4*R*x + 2*l^2 - 2*m^2 + r^2 + 2*x^2 + 2*y^2 + 2*z^2 - 2^(1/2)*R*r + 2*r*y*sin(pi/4 + 2*alpha) + 2^(1/2)*r*x - 2^(1/2)*r*y - 2^(1/2)*R*r*cos(2*alpha) + 2^(1/2)*R*r*sin(2*alpha) + 2^(1/2)*r*x*cos(2*alpha) - 2^(1/2)*r*x*sin(2*alpha)))
-2*atan((2*cos(alpha/2)^4*(2*R*l - 2*l*x + ((16*l^2*(2*R - 2*x - 2^(1/2)*r*cos(alpha)^2 + (2^(1/2)*r*sin(2*alpha))/2)^2)/(cos(alpha) + 1)^4 - ((2*R^2 - 4*l*z - r^2*sin(2*alpha) - 4*R*x + 2*l^2 - 2*m^2 + r^2 + 2*x^2 + 2*y^2 + 2*z^2 - 2^(1/2)*R*r + 2*r*y*sin(pi/4 + 2*alpha) + 2^(1/2)*r*x - 2^(1/2)*r*y - 2^(1/2)*R*r*cos(2*alpha) + 2^(1/2)*R*r*sin(2*alpha) + 2^(1/2)*r*x*cos(2*alpha) - 2^(1/2)*r*x*sin(2*alpha))*(4*l*z - 4*R*x - r^2*sin(2*alpha) + 2*R^2 + 2*l^2 - 2*m^2 + r^2 + 2*x^2 + 2*y^2 + 2*z^2 - 2^(1/2)*R*r + 2*r*y*sin(pi/4 + 2*alpha) + 2^(1/2)*r*x - 2^(1/2)*r*y - 2^(1/2)*R*r*cos(2*alpha) + 2^(1/2)*R*r*sin(2*alpha) + 2^(1/2)*r*x*cos(2*alpha) - 2^(1/2)*r*x*sin(2*alpha)))/(4*cos(alpha/2)^8))^(1/2) - 2^(1/2)*l*r + 4*R*l*tan(alpha/2)^2 + 2*R*l*tan(alpha/2)^4 - 4*l*x*tan(alpha/2)^2 - 2*l*x*tan(alpha/2)^4 + 2*2^(1/2)*l*r*tan(alpha/2)^2 - 2*2^(1/2)*l*r*tan(alpha/2)^3 - 2^(1/2)*l*r*tan(alpha/2)^4 + 2*2^(1/2)*l*r*tan(alpha/2)))/(2*R^2 - 4*l*z - r^2*sin(2*alpha) - 4*R*x + 2*l^2 - 2*m^2 + r^2 + 2*x^2 + 2*y^2 + 2*z^2 - 2^(1/2)*R*r + 2*r*y*sin(pi/4 + 2*alpha) + 2^(1/2)*r*x - 2^(1/2)*r*y - 2^(1/2)*R*r*cos(2*alpha) + 2^(1/2)*R*r*sin(2*alpha) + 2^(1/2)*r*x*cos(2*alpha) - 2^(1/2)*r*x*sin(2*alpha)))
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