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Thread Subject:
"Explicit solution could not be found" in solving system of equations

Subject: "Explicit solution could not be found" in solving system of equations

From: haunteagle iœ

Date: 3 Apr, 2009 08:26:02

Message: 1 of 5

>> solve('(a * cos(b) - 0.000)^2 + (a * sin(b) - 0.450)^2 = (a + c * t1)^2', '(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.450)^2 = (a + c * t2)^2','(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.000)^2 = (a + c * t4)^2','a','b','c')

I was trying to solve the equations above when I met the problem says:

"Warning: Explicit solution could not be found.
>In solve at 170"

I have checked my equations for many times and I also search the help files, but I cannot find out a solution.

Please help revise the equations and give an explanation why.

Any help will be appreciated.

Subject: "Explicit solution could not be found" in solving system of equations

From: Roger Stafford

Date: 3 Apr, 2009 16:28:01

Message: 2 of 5

"haunteagle iœ" <haunteagle@gmail.com> wrote in message <gr4h6q$rd9$1@fred.mathworks.com>...
> >> solve('(a * cos(b) - 0.000)^2 + (a * sin(b) - 0.450)^2 = (a + c * t1)^2', '(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.450)^2 = (a + c * t2)^2','(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.000)^2 = (a + c * t4)^2','a','b','c')
> .......
> "Warning: Explicit solution could not be found.
> .......

  With some fairly messy algebraic manipulations, the 'a' and 'b' unknowns in your equations can be eliminated, leaving a single polynomial equation in the 'c' unknown of (I believe) the tenth order. Perhaps that is why 'solve' gave up without giving you an explicit solution, since explicit solutions of such polynomials equations cannot be given in terms of elementary functions.

  However you could use the 'roots' function to obtain numerical solutions
for specific values of 't1', 't2', and 't4', and this would avoid having to use the iterative methods of the Optimization Toolbox.

Roger Stafford

Subject: "Explicit solution could not be found" in solving system of equations

From: Walter Roberson

Date: 3 Apr, 2009 20:01:24

Message: 3 of 5

Roger Stafford wrote:
> "haunteagle iœ" <haunteagle@gmail.com> wrote in message <gr4h6q$rd9$1@fred.mathworks.com>...
>>>> solve('(a * cos(b) - 0.000)^2 + (a * sin(b) - 0.450)^2 = (a + c * t1)^2', '(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.450)^2 = (a + c * t2)^2','(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.000)^2 = (a + c * t4)^2','a','b','c')
>> .......
>> "Warning: Explicit solution could not be found.
>> .......
>
> With some fairly messy algebraic manipulations, the 'a' and 'b' unknowns in your equations
> can be eliminated, leaving a single polynomial equation in the 'c' unknown of (I believe)
> the tenth order.

Maple 13 beta resolves everything down to expressions in quartics -- which there are explicit roots
for (some of which might be complex)

#1)

[a = 3/40*13^(1/2), b = arctan(3/2), c = 0]

#2)

[a = -3/40*13^(1/2), b = arctan(3/2)-Pi, c = 0]

#3)

t3 = (t4 ^ 2);
t5 = (t1 ^ 2);
t6 = (t2 ^ 2);
t7 = t3 + t5 - t6;
t8 = sqrt(0.6e1);
t10 = (t5 ^ 2);
t12 = (t5 * t1);
t13 = (t12 * t2);
t15 = (t6 * t5);
t22 = (t6 * t2);
t23 = (t22 * t1);
t28 = (t6 * t1);
t33 = (t3 ^ 2);
t35 = (t6 ^ 2);
t37 = (t3 * t4);
t42 = (t33 ^ 2);
t44 = (t35 ^ 2);
t52 = (t35 * t5);
t55 = (t10 * t1);
t59 = (t33 * t4);
t63 = (t10 * t5);
t67 = (t33 * t3);
t74 = (t22 * t12);
t77 = (t6 * t12);
t83 = (t22 * t37);
t86 = (t5 * t33);
t89 = (t10 * t6);
t102 = 9 * t42 + 9 * t44 - 784 * t10 * t2 * t37 + 372 * t10 * t22 * t4 + 565 * t52 * t3 + 334 * t55 * t3 * t2 + 464 * t59 * t2 * t5 + 68 * t63 * t4 * t2 + 198 * t67 * t1 * t2 - 272 * t6 * t55 * t4 - 828 * t74 * t3 + 1640 * t77 * t37 - 792 * t6 * t59 * t1 - 568 * t83 * t5 - 200 * t86 * t6 + 125 * t89 * t3 - 784 * t13 * t33 + 1152 * t23 * t33 - 648 * t35 * t37 * t1 - 128 * t35 * t12 * t4;
t103 = t35 * t2;
t110 = t35 * t6;
t147 = t35 * t22;
t152 = t10 ^ 2;
t154 = 100 * t22 * t55 + 360 * t22 * t59 - 36 * t10 * t12 * t2 - 36 * t2 * t33 * t37 + 100 * t103 * t12 + 20 * t110 * t5 + 360 * t103 * t37 - 45 * t110 * t3 - 36 * t147 * t4 - 36 * t147 * t1 + 9 * t152;
t157 = sqrt((t102 - 18 * t103 * t3 * t1 - 148 * t103 * t5 * t4 + 144 * t110 * t1 * t4 - 133 * t63 * t3 + 392 * t10 * t33 + 20 * t63 * t6 - 133 * t5 * t67 - 45 * t67 * t6 - 186 * t35 * t10 - 576 * t35 * t33 + t154));
t158 = (3 * t10) - (6 * t13) + (6 * t15) + (6 * t5 * t3) - (6 * t2 * t4 * t5) - (6 * t23) - (6 * t1 * t2 * t3) + (12 * t28 * t4) + (6 * t6 * t3) + (3 * t33) + (3 * t35) - (6 * t2 * t37) - (6 * t22 * t4) + t157;
t190 = -8 * t10 * t4 * t2 + 13 * t10 * t3 + 4 * t89 + 8 * t77 * t4 + 8 * t12 * t3 * t2 - 8 * t74 - 26 * t12 * t37 + 4 * t52 - 26 * t15 * t3 + 13 * t86 + 18 * t37 * t5 * t2 + 18 * t28 * t37 - 18 * t33 * t1 * t2 - 18 * t83 + 9 * t33 * t6 + 9 * t35 * t3;
t191 = 1 / t190;
t193 = sqrt(t158 * t191);
a = -0.3e1 / 0.20e2 * t7 * t8 * t193 / (2 * t4 + 2 * t1 - 2 * t2);
t203 = t3 * t1;
t204 = t5 * t4;
t212 = 0.1e1 / t193;
b = atan2((-(81 * t4) - (81 * t1) + (81 * t2) + 0.27e2 / 0.200e3 * (-400 * t203 + 400 * t28 + 400 * t204 - 400 * t5 * t2) * t158 * t191) * t212 / t7 / 0.180e3, (-(9 * t4) - (9 * t1) + (9 * t2) + 0.27e2 / 0.200e3 * (-100 * t204 + 100 * t6 * t4 + 100 * t203 - 100 * t3 * t2) * t158 * t191) * t212 / t7 / 0.30e2);
c = 0.3e1 / 0.20e2 * t8 * t193;


#4)

t3 = (t4 ^ 2);
t5 = (t1 ^ 2);
t6 = (t2 ^ 2);
t7 = t3 + t5 - t6;
t8 = sqrt(0.6e1);
t10 = (t5 ^ 2);
t12 = (t5 * t1);
t13 = (t12 * t2);
t15 = (t6 * t5);
t22 = (t6 * t2);
t23 = (t22 * t1);
t28 = (t6 * t1);
t33 = (t3 ^ 2);
t35 = (t6 ^ 2);
t37 = (t3 * t4);
t42 = (t10 * t5);
t49 = (t33 * t3);
t58 = (t10 * t1);
t67 = (t35 * t5);
t73 = (t33 * t4);
t86 = (t22 * t12);
t89 = (t6 * t12);
t95 = (t22 * t37);
t98 = -133 * t42 * t3 + 392 * t10 * t33 + 20 * t42 * t6 - 133 * t5 * t49 - 45 * t49 * t6 - 186 * t35 * t10 - 576 * t35 * t33 + 100 * t22 * t58 - 784 * t10 * t2 * t37 + 372 * t10 * t22 * t4 + 565 * t67 * t3 + 334 * t58 * t3 * t2 + 464 * t73 * t2 * t5 + 68 * t42 * t4 * t2 + 198 * t49 * t1 * t2 - 272 * t6 * t58 * t4 - 828 * t86 * t3 + 1640 * t89 * t37 - 792 * t6 * t73 * t1 - 568 * t95 * t5;
t99 = t5 * t33;
t102 = t10 * t6;
t113 = t35 * t2;
t116 = t35 * t6;
t123 = t35 * t22;
t129 = t10 ^ 2;
t131 = t33 ^ 2;
t133 = t35 ^ 2;
t154 = -36 * t123 * t1 + 9 * t129 + 9 * t131 + 9 * t133 - 784 * t13 * t33 + 1152 * t23 * t33 - 648 * t35 * t37 * t1 - 128 * t35 * t12 * t4 - 18 * t113 * t3 * t1 - 148 * t113 * t5 * t4 + 144 * t116 * t1 * t4;
t157 = sqrt((t98 - 200 * t99 * t6 + 125 * t102 * t3 + 360 * t22 * t73 - 36 * t10 * t12 * t2 - 36 * t2 * t33 * t37 + 100 * t113 * t12 + 20 * t116 * t5 + 360 * t113 * t37 - 45 * t116 * t3 - 36 * t123 * t4 + t154));
t158 = (3 * t10) - (6 * t13) + (6 * t15) + (6 * t5 * t3) - (6 * t2 * t4 * t5) - (6 * t23) - (6 * t1 * t2 * t3) + (12 * t28 * t4) + (6 * t6 * t3) + (3 * t33) + (3 * t35) - (6 * t2 * t37) - (6 * t22 * t4) + t157;
t190 = -8 * t10 * t4 * t2 + 13 * t10 * t3 + 4 * t102 + 8 * t89 * t4 + 8 * t12 * t3 * t2 - 8 * t86 - 26 * t12 * t37 + 4 * t67 - 26 * t15 * t3 + 13 * t99 + 18 * t37 * t5 * t2 + 18 * t28 * t37 - 18 * t33 * t1 * t2 - 18 * t95 + 9 * t33 * t6 + 9 * t35 * t3;
t191 = 1 / t190;
t193 = sqrt(t158 * t191);
a = 0.3e1 / 0.20e2 * t7 * t8 * t193 / (2 * t4 + 2 * t1 - 2 * t2);
t202 = t3 * t1;
t203 = t5 * t4;
t211 = 0.1e1 / t193;
b = atan2(-(-(81 * t4) - (81 * t1) + (81 * t2) + 0.27e2 / 0.200e3 * (-400 * t202 + 400 * t28 + 400 * t203 - 400 * t5 * t2) * t158 * t191) * t211 / t7 / 0.162e3, -(-(9 * t4) - (9 * t1) + (9 * t2) + 0.27e2 / 0.200e3 * (-100 * t203 + 100 * t6 * t4 + 100 * t202 - 100 * t3 * t2) * t158 * t191) * t211 / t7 / 0.27e2);
c = -0.3e1 / 0.20e2 * t8 * t193;


#5)

t3 = (t4 ^ 2);
t5 = (t1 ^ 2);
t6 = (t2 ^ 2);
t7 = t3 + t5 - t6;
t8 = (t5 ^ 2);
t10 = (t5 * t1);
t11 = (t10 * t2);
t13 = (t6 * t5);
t20 = (t6 * t2);
t21 = (t20 * t1);
t26 = (t6 * t1);
t31 = (t3 ^ 2);
t33 = (t6 ^ 2);
t35 = (t3 * t4);
t46 = (t5 * t33);
t49 = (t8 * t1);
t53 = (t31 * t4);
t57 = (t8 * t5);
t61 = (t31 * t3);
t68 = (t20 * t10);
t71 = (t10 * t6);
t77 = (t20 * t35);
t80 = (t5 * t31);
t83 = (t8 * t6);
t96 = (t33 * t2);
t103 = -784 * t8 * t2 * t35 + 372 * t8 * t20 * t4 + 565 * t46 * t3 + 334 * t49 * t3 * t2 + 464 * t53 * t2 * t5 + 68 * t57 * t4 * t2 + 198 * t61 * t1 * t2 - 272 * t6 * t49 * t4 - 828 * t68 * t3 + 1640 * t71 * t35 - 792 * t6 * t53 * t1 - 568 * t77 * t5 - 200 * t80 * t6 + 125 * t83 * t3 - 784 * t11 * t31 + 1152 * t21 * t31 - 648 * t33 * t35 * t1 - 128 * t33 * t10 * t4 - 18 * t96 * t3 * t1 - 148 * t96 * t5 * t4;
t104 = t33 * t6;
t108 = t8 ^ 2;
t110 = t31 ^ 2;
t112 = t33 ^ 2;
t147 = t33 * t20;
t152 = -576 * t33 * t31 + 100 * t20 * t49 + 360 * t20 * t53 - 36 * t8 * t10 * t2 - 36 * t2 * t31 * t35 + 100 * t96 * t10 + 20 * t104 * t5 + 360 * t96 * t35 - 45 * t104 * t3 - 36 * t147 * t4 - 36 * t147 * t1;
t155 = sqrt((t103 + 144 * t104 * t1 * t4 + 9 * t108 + 9 * t110 + 9 * t112 - 133 * t57 * t3 + 392 * t8 * t31 + 20 * t57 * t6 - 133 * t5 * t61 - 45 * t61 * t6 - 186 * t33 * t8 + t152));
t157 = -(18 * t8) + (36 * t11) - (36 * t13) - (36 * t5 * t3) + (36 * t2 * t4 * t5) + (36 * t21) + (36 * t1 * t2 * t3) - (72 * t26 * t4) - (36 * t6 * t3) - (18 * t31) - (18 * t33) + (36 * t35 * t2) + (36 * t20 * t4) + 0.6e1 * t155;
t189 = -8 * t8 * t4 * t2 + 13 * t8 * t3 + 4 * t83 + 8 * t71 * t4 + 8 * t10 * t3 * t2 - 8 * t68 - 26 * t35 * t10 + 4 * t46 - 26 * t13 * t3 + 13 * t80 + 18 * t35 * t5 * t2 + 18 * t26 * t35 - 18 * t31 * t1 * t2 - 18 * t77 + 9 * t31 * t6 + 9 * t33 * t3;
t190 = 1 / t189;
t192 = sqrt(-t157 * t190);
a = -0.3e1 / 0.20e2 * t7 * t192 / (2 * t4 + 2 * t1 - 2 * t2);
t202 = t3 * t1;
t203 = t5 * t4;
t211 = 0.1e1 / t192;
b = atan2((-(81 * t4) - (81 * t1) + (81 * t2) - 0.9e1 / 0.400e3 * (-400 * t202 + 400 * t26 + 400 * t203 - 400 * t5 * t2) * t157 * t190) * t211 / t7 / 0.180e3, (-(9 * t4) - (9 * t1) + (9 * t2) - 0.9e1 / 0.400e3 * (-100 * t203 + 100 * t6 * t4 + 100 * t202 - 100 * t3 * t2) * t157 * t190) * t211 / t7 / 0.30e2);
c = 0.3e1 / 0.20e2 * t192;


#6)

t3 = (t4 ^ 2);
t5 = (t1 ^ 2);
t6 = (t2 ^ 2);
t7 = t3 + t5 - t6;
t8 = (t5 ^ 2);
t10 = (t5 * t1);
t11 = (t10 * t2);
t13 = (t6 * t5);
t20 = (t6 * t2);
t21 = (t20 * t1);
t26 = (t6 * t1);
t31 = (t3 ^ 2);
t33 = (t6 ^ 2);
t35 = (t3 * t4);
t46 = (t5 * t33);
t49 = (t8 * t1);
t53 = (t31 * t4);
t57 = (t8 * t5);
t61 = (t31 * t3);
t68 = (t20 * t10);
t71 = (t10 * t6);
t77 = (t20 * t35);
t80 = (t5 * t31);
t83 = (t8 * t6);
t96 = (t33 * t2);
t103 = -784 * t8 * t2 * t35 + 372 * t8 * t20 * t4 + 565 * t46 * t3 + 334 * t49 * t3 * t2 + 464 * t53 * t2 * t5 + 68 * t57 * t4 * t2 + 198 * t61 * t1 * t2 - 272 * t6 * t49 * t4 - 828 * t68 * t3 + 1640 * t71 * t35 - 792 * t6 * t53 * t1 - 568 * t77 * t5 - 200 * t80 * t6 + 125 * t83 * t3 - 784 * t11 * t31 + 1152 * t21 * t31 - 648 * t33 * t35 * t1 - 128 * t33 * t10 * t4 - 18 * t96 * t3 * t1 - 148 * t96 * t5 * t4;
t104 = t33 * t6;
t141 = t33 * t20;
t146 = t8 ^ 2;
t148 = t31 ^ 2;
t150 = t33 ^ 2;
t152 = -36 * t8 * t10 * t2 - 36 * t2 * t31 * t35 + 100 * t96 * t10 + 20 * t104 * t5 + 360 * t96 * t35 - 45 * t104 * t3 - 36 * t141 * t4 - 36 * t141 * t1 + 9 * t146 + 9 * t148 + 9 * t150;
t155 = sqrt((t103 + 144 * t104 * t1 * t4 - 133 * t57 * t3 + 392 * t8 * t31 + 20 * t57 * t6 - 133 * t5 * t61 - 45 * t61 * t6 - 186 * t33 * t8 - 576 * t33 * t31 + 100 * t20 * t49 + 360 * t20 * t53 + t152));
t157 = -(18 * t8) + (36 * t11) - (36 * t13) - (36 * t5 * t3) + (36 * t2 * t4 * t5) + (36 * t21) + (36 * t1 * t2 * t3) - (72 * t26 * t4) - (36 * t6 * t3) - (18 * t31) - (18 * t33) + (36 * t35 * t2) + (36 * t20 * t4) + 0.6e1 * t155;
t189 = -8 * t8 * t4 * t2 + 13 * t8 * t3 + 4 * t83 + 8 * t71 * t4 + 8 * t10 * t3 * t2 - 8 * t68 - 26 * t35 * t10 + 4 * t46 - 26 * t13 * t3 + 13 * t80 + 18 * t35 * t5 * t2 + 18 * t26 * t35 - 18 * t31 * t1 * t2 - 18 * t77 + 9 * t31 * t6 + 9 * t33 * t3;
t190 = 1 / t189;
t192 = sqrt(-t157 * t190);
a = 0.3e1 / 0.20e2 * t7 * t192 / (2 * t4 + 2 * t1 - 2 * t2);
t201 = t3 * t1;
t202 = t5 * t4;
t210 = 0.1e1 / t192;
b = atan2(-(-(81 * t4) - (81 * t1) + (81 * t2) - 0.9e1 / 0.400e3 * (-400 * t201 + 400 * t26 + 400 * t202 - 400 * t5 * t2) * t157 * t190) * t210 / t7 / 0.27e2, -0.2e1 / 0.9e1 * (-(9 * t4) - (9 * t1) + (9 * t2) - 0.9e1 / 0.400e3 * (-100 * t202 + 100 * t6 * t4 + 100 * t201 - 100 * t3 * t2) * t157 * t190) * t210 / t7);
c = -0.3e1 / 0.20e2 * t192;


Or if you happen to prefer Maple notation, the list as a whole is:

[[a = 3/40*13^(1/2), b = arctan(3/2), c = 0],

[a = -3/40*13^(1/2), b = arctan(3/2)-Pi, c = 0],

[a = -3/20*(t4^2+t1^2-t2^2)*6^(1/2)*((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9
*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(2*t4+2*t1-2*t2), b = arctan((-81*t4-81*t1+81*t2+27/200*(-400*t4^2*t1+400*t2^2*t1+400*t1^2*t4-400*t1^2*t2)*(3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t
2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*
t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(180*t4^2+180*t1^2-180*t2^2),(-9*t4-9*t1+9*t2+27/200*(-100*t1^2*t4+100*t2^2*t4+100*t4^2*t1-100*t4^2*t2)*(3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t
2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4
^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(30*t4^2+30*t1^2-30*t2^2)), c = 3/20*6^(1/2)*((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2
^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)],

[a = 3/20*(t4^2+t1^2-t2^2)*6^(1/2)*((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*
t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(2*t4+2*t1-2*t2), b = arctan(-10/9*(-81*t4-81*t1+81*t2+27/200*(-400*t4^2*t1+400*t2^2*t1+400*t1^2*t4-400*t1^2*t2)*(3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^
3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^
2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(180*t4^2+180*t1^2-180*t2^2),-10/9*(-9*t4-9*t1+9*t2+27/200*(-100*t1^2*t4+100*t2^2*t4+100*t4^2*t1-100*t4^2*t2)*(3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-
8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2
+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(30*t4^2+30*t1^2-30*t2^2)), c = -3/20*6^(1/2)*((3*t1^4-6*t1^3*t2+6*t2^2*t1^2+6*t1^2*t4^2-6*t2*t4*t1^2-6*t2^3*t1-6*t1*t2*t4^2+12*t2^2*t1*t4+6*t2^2*t4^2+3*t4^4+3*t2^4-6*t2*t4^3-6*t2^3*t4+(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6
*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)],

[a = -3/20*(t4^2+t1^2-t2^2)*(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^
3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(2*t4+2*t1-2*t2), b = arctan((-81*t4-81*t1+81*t2-9/400*(-400*t4^2*t1+400*t2^2*t1+400*t1^2*t4-400*t1^2*t2)*(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3
*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*
t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(180*t4^2+180*t1^2-180*t2^2),(-9*t4-9*t1+9*t2-9/400*(-100*t1^2*t4+100*t2^2*t4+100*t4^2*t1-100*t4^2*t2)*(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^
7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36
*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(30*t4^2+30*t1^2-30*t2^2)), c = 3/20*(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2
^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)],

[a = 3/20*(t4^2+t1^2-t2^2)*(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3
*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(2*t4+2*t1-2*t2), b = arctan(-20/3*(-81*t4-81*t1+81*t2-9/400*(-400*t4^2*t1+400*t2^2*t1+400*t1^2*t4-400*t1^2*t2)*(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8
*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*
t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(180*t4^2+180*t1^2-180*t2^2),-20/3*(-9*t4-9*t1+9*t2-9/400*(-100*t1^2*t4+100*t2^2*t4+100*t4^2*t1-100*t4^2*t2)*(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*
t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))/(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^
2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)/(30*t4^2+30*t1^2-30*t2^2)), c = -3/20*(-(-18*t1^4+36*t1^3*t2-36*t2^2*t1^2-36*t1^2*t4^2+36*t2*t4*t1^2+36*t2^3*t1+36*t1*t2*t4^2-72*t2^2*t1*t4-36*t2^2*t4^2-18*t4^4-18*t2^4+36*t2*t4^3+36*t2^3*t4+6*(-784*t1^4*t2*t4^3+372*t1^4*t2^3*t4+565*t2^4*t1^2*t4^2+334*t1^5*t4^2*t2+464*t4^5*t2*t1^2+68*t1^6*t4*t2+198*t4^6*t1*t2-272*t2^2*t1^5*t4-828*t2^3*t1^3*t4^2+1640*t2^2*t1^3*t4^3-792*t2^2*t4^5*t1-568*t2^3*t4^3*t1^2-200*t1^2*t4^4*t2^2+125*t1^4*t2^2*t4^2-133*t1^6*t4^2+392*t1^4*t4^4+20*t1^6*t2^2-133*t1^2*t4^6-45*t4^6*t2^2-186*t2^4*t1^4-576*t2^4*t4^4+100*t2^3*t1^5+360*t2^3*t4^5-36*t1^7*t2-36*t2*t4^7-784*t1^3*t2*t4^4+1152*t2^3*t1*t4^4+9*t1^8+9*t4^8+100*t2^5*t1^3+20*t2^6*t1^2-648*t2^4*t4^3*t1-128*t2^4*t1^3*t4-18*t2^5*t4
^2*t1-148*t2^5*t1^2*t4+360*t2^5*t4^3-45*t2^6*t4^2+9*t2^8-36*t2^7*t4+144*t2^6*t1*t4-36*t2^7*t1)^(1/2))/(-8*t1^4*t4*t2+13*t1^4*t4^2+4*t1^4*t2^2+8*t2^2*t1^3*t4+8*t1^3*t4^2*t2-8*t2^3*t1^3-26*t1^3*t4^3+4*t2^4*t1^2-26*t1^2*t2^2*t4^2+13*t4^4*t1^2+18*t4^3*t1^2*t2+18*t2^2*t1*t4^3-18*t4^4*t1*t2-18*t2^3*t4^3+9*t4^4*t2^2+9*t2^4*t4^2))^(1/2)]]


Note: I did not attempt to verify this solution!!

Subject: "Explicit solution could not be found" in solving system of equations

From: Roger Stafford

Date: 4 Apr, 2009 02:25:03

Message: 4 of 5

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <gr5deh$kgt$1@fred.mathworks.com>...
> "haunteagle iœ" <haunteagle@gmail.com> wrote in message <gr4h6q$rd9$1@fred.mathworks.com>...
> > >> solve('(a * cos(b) - 0.000)^2 + (a * sin(b) - 0.450)^2 = (a + c * t1)^2', '(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.450)^2 = (a + c * t2)^2','(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.000)^2 = (a + c * t4)^2','a','b','c')
> > .......
> > "Warning: Explicit solution could not be found.
> > .......
>
> With some fairly messy algebraic manipulations, the 'a' and 'b' unknowns in your equations can be eliminated, leaving a single polynomial equation in the 'c' unknown of (I believe) the tenth order. Perhaps that is why 'solve' gave up without giving you an explicit solution, since explicit solutions of such polynomials equations cannot be given in terms of elementary functions.
>
> However you could use the 'roots' function to obtain numerical solutions
> for specific values of 't1', 't2', and 't4', and this would avoid having to use the iterative methods of the Optimization Toolbox.
>
> Roger Stafford

  Yes, as it turns out there is a quartic, as Walter has stated. The unknown 'c' is a root of a quartic with the odd powers of 'c' missing, so it is actually a quadratic equation in c^2 for which it is easy to find explicit roots. 'c' must satisfy:

 k4*c^4 + k2*c^2 + k0 = 0

where

 k4 = t4^2*(t1-t2)^2*(t1-t4+t2)^2*p^2+t1^2*(t4-t2)^2*(t4-t1+t2)^2*q^2
 k2 = -(t1^2+t4^2+t2^2-2*t1*t2)*(t1^2+t4^2+t2^2-2*t4*t2)*p^2*q^2
 k0 = (t1+t4-t2)^2*(p^2+q^2)*p^2*q^2

and p = 0.450 and q = 0.300. Thus it is not quite as frightening as it appears at first glance in Walter's solutions.

  From 'c' it is then possible to evaluate 'b' and 'a'.

Roger Stafford

Subject: "Explicit solution could not be found" in solving system of equations

From: Roger Stafford

Date: 4 Apr, 2009 21:42:02

Message: 5 of 5

"haunteagle iœ" <haunteagle@gmail.com> wrote in message <gr4h6q$rd9$1@fred.mathworks.com>...
> >> solve('(a * cos(b) - 0.000)^2 + (a * sin(b) - 0.450)^2 = (a + c * t1)^2', '(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.450)^2 = (a + c * t2)^2','(a * cos(b) - 0.300)^2 + (a * sin(b) - 0.000)^2 = (a + c * t4)^2','a','b','c')
> ......

  I finally got around to finishing that problem for you, Haunteagle. Here is the complete solution.

  We are given t1, t2, and t4. Define p = 0.450 and q = 0.300 . (We assume that your 0.000 is exactly zero.) Then, as I said before, c is a root of a quadratic in c^2:

 k2 = t4^2*(t1-t2)^2*(t1-t4+t2)^2*p^2+t1^2*(t4-t2)^2*(t4-t1+t2)^2*q^2;
 k1 = -(t1^2+t4^2+t2^2-2*t1*t2)*(t1^2+t4^2+t2^2-2*t4*t2)*p^2*q^2;
 k0 = (t1+t4-t2)^2*(p^2+q^2)*p^2*q^2;
 c = sqrt(roots([k2,k1,k0]));
 c = [c;-c]; % Get all four roots

  For those roots in c that are real, the quantities a and b can be determined as follows:

 a = (t4^2+t1^2-t2^2)*c/(t2-t1-t4)/2;
 sb = ((t2-t1-t4)*p^2+t1*(t2-t4)*(t2+t4-t1)*c^2)/(t1^2+t4^2-t2^2)/p/c;
 cb = ((t2-t1-t4)*q^2+t4*(t2-t1)*(t2+t1-t4)*c^2)/(t1^2+t4^2-t2^2)/q/c;
 b = atan2(sb,cb);

  These equations were determined using the reasoning I described in my first article above. The quantity a is first eliminated from the three original equations, yielding two equations in b and c. Then sb = sin(b) and cb = cos(b) are obtained in terms of c, solving the two equations which are linear in sb and cb. Then the condition sb^2+cb^2 = 1 is applied to this, with a resulting quartic equation in c. Next the equations for sb and cb are evaluated in terms of c, getting values for sb and cb. Using atan2 on these gives b lying in [-pi,pi]. Finally the expression for a is obtained from one of the original equations using these expressions for sb and cb in terms of c.

  As you have seem from Walter's results, it is better in this case not to use 'solve' to find all three unknowns in only one step, since that results in a very messy answer.

Roger Stafford

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