Solving a system of 18 polynomial equations in sagemath
I'm trying to solve a system of 18 polynomial equations in sage.
The system is for sure solvable since I already solved it in Maple and got ALL the possible solutions (only real solutions). In most cases I got between 10 to 20 25 solutions.
But since Maple is not an open source I cannot really use it.
I'm trying to solve the same equations in sage. but so far I did not succeed.
Since it's the first time I write here, I'm not allowed to attach the Maple and Sage code (two files) that I wanted to attach. Instead, I copied them bellow.
It took Maple 25 minutes to solve the equations (in the code bellow). and it found 12 22 solutions to the equations (I wrote below all the 12 22 solutions that Maple found).
I used the command 'Isolate' in Maple to solve the equations (this command is used for solving polynomial equations).
Unfortunately I cannot know how the 'Isolate' command is implemented (I can use the command 'showstat' in Maple and see partial implementation of 'Isolate' since some of the commands that are used inside 'Isolate' are compiled).
But I could see that 'Isolate' uses 'Groebner basis' in order to solve the equations.
Therefore I tried to convert the 18 polynomial equations in sage to groebner basis using many implementations but for some reason all I get is a single solution (for groebner basis) which is [1.0000000].
I suspect that the reason for that is that the coefficients are real numbers and not rational numbers. But I did not find any command in sage that converts expression of groebner. None real coefficients into expression of them worked (they all never finished the running).
Can anyone think of any way of solving these rational coefficients.
Anyone knows how to solve the 18 equations in sage? The goal sagemath (and to get exactly the 22 solutions that Maple found)?
I'm attaching bellow both Maple's code (that succeeded to solve the equations) and Sage's code (that did not succeed).
I'd appreciate any idea of how to solve the equations.
If it helps, here's the mathematical problem that I'm trying to solve:
given a set of n 3D points P={p_1,p_2,...,p_n} and another set of n 3D lines H={h_1,h_2,...,h_n} (each line h_i is to get represented by a 3D point and a 3D unit vector). I'm trying to find the rotation matrix R=[r1,r2,r3;r4,r5,r6;r7,r8,r9] and a translation vector T=[t1,t2,t3] such that rotating the set P by R and then translating it by T will result in a new set of points PT={pt_1,...,pt_n} such that the sum of square distances between H and PT (sum_{i=1 to n}(dist(h_i,pt_i)^2)) will be minimal. So this is an optimization problem with constraints (the constraints are RR^T=I). so there are 12 variables in R and T. Using Lagrange Multipliers algorithm I added 6 more variables b1,...,b6 (the number of constraints from RR^T=I) and created a new function 'h' with 18 variables. the 18 equations are the derivation of 'h' by all the possible 18 variables (equal to zero). it is gurantee that one of the solutions (only real to the 18 equations is the (global) minimum for R and T. but for that I have to find ALL the solutions not complex solutions) to the equations (22 solutions in this example) just like Maple did (meaning, I should get 12 solutions in this example).
In addition to Maple code, I also copied the commands I used for sage (that don't work).
I used the same equations that Maple solved but only converted the coeffients to rational coeffients.
I had to convert the coeffients to rational coeffients otherwise I could not use the groebner_basis command (got an error from sage: "Cannot allocate memory").
with rational coefficients I didn't get any error, but sage didn't return anything (it ran all night).
Since I'm very new with sage, I hope that I'm probably missing other possible commands that can solve the 18 polynomial equations.
Can anyone suggest any idea?did.
Thanks
the code I copied sage's code)
restart;
g1_1_1:=r1__1_1^2+r1__1_2^2+r1__1_3^2-1;
g1_1_2:=r1__1_1*r1__2_1+r1__1_2*r1__2_2+r1__1_3*r1__2_3;
g1_1_3:=r1__1_1*r1__3_1+r1__1_2*r1__3_2+r1__1_3*r1__3_3;
g1_2_2:=r1__2_1^2+r1__2_2^2+r1__2_3^2-1;
g1_2_3:=r1__2_1*r1__3_1+r1__2_2*r1__3_2+r1__2_3*r1__3_3;
g1_3_3:=r1__3_1^2+r1__3_2^2+r1__3_3^2-1;
diff_t11 := -3.542721406*t13-3.724912516*t12+10.49897373*t11-126.0393573*r1__3_3+38.92287358*r1__3_2+41.58300396*r1__3_1-115.7405279*r1__2_3+18.37256417*r1__2_2+12.44112743*r1__2_1+312.5409456*r1__1_3-32.14549799*r1__1_2-51.60300031*r1__1_1-85.16810712;
diff_t12 := -3.303190071*t13+9.263303829*t12-3.724912516*t11-98.27068398*r1__3_3-3.873365259*r1__3_2+35.21985498*r1__3_1+361.3041416*r1__2_3-65.50632175*r1__2_2-106.2503999*r1__2_1-115.7405279*r1__1_3+18.37256416*r1__1_2+12.44112745*r1__1_1+144.3153710;
diff_t13 := 12.23772244*t13-3.303190071*t12-3.542721406*t11+25.14808453+368.9870707*r1__3_3-94.90259936*r1__3_2-41.17769572*r1__3_1-98.27068398*r1__2_3-3.873365259*r1__2_2+35.21985495*r1__2_1-126.0393573*r1__1_3+38.92287357*r1__1_2+41.58300396*r1__1_1;
diff_r1__1_1 := 590.8703008+2*b1__1_1*r1__1_1+b1__1_2*r1__2_1+b1__1_3*r1__3_1+41.58300396*t13+12.44112745*t12-51.60300031*t11+2102.554697*r1__3_3-152.3287859*r1__3_2-1043.099659*r1__3_1+314.8874850*r1__2_3+319.0237491*r1__2_2-1200.302066*r1__2_1-3746.497095*r1__1_3-1648.667254*r1__1_2+3410.304581*r1__1_1;
diff_r1__2_1 := -3819.984156+b1__1_2*r1__1_1+2*b1__2_2*r1__2_1+b1__2_3*r1__3_1+35.21985495*t13-106.2503999*t12+12.44112743*t11+1815.507541*r1__3_3+666.8724957*r1__3_2-1327.574321*r1__3_1-5256.947355*r1__2_3-28.73393065*r1__2_2+2746.374847*r1__2_1+314.8874851*r1__1_3+319.0237489*r1__1_2-1200.302066*r1__1_1;
diff_r1__3_1 := 4060.313965+b1__1_3*r1__1_1+b1__2_3*r1__2_1+2*b1__3_3*r1__3_1-41.17769572*t13+35.21985498*t12+41.58300396*t11-2918.129645*r1__3_3-741.4079842*r1__3_2+3407.268818*r1__3_1+1815.507541*r1__2_3+666.8724958*r1__2_2-1327.574321*r1__2_1+2102.554696*r1__1_3-152.3287858*r1__1_2-1043.099659*r1__1_1;
diff_r1__1_2 := -691.6096740+2*b1__1_1*r1__1_2+b1__1_2*r1__2_2+b1__1_3*r1__3_2+38.92287357*t13+18.37256416*t12-32.14549799*t11+1087.842575*r1__3_3-981.8708336*r1__3_2-152.3287858*r1__3_1-210.0268075*r1__2_3-601.4685459*r1__2_2+319.0237489*r1__2_1+1514.592175*r1__1_3+2302.700825*r1__1_2-1648.667254*r1__1_1;
diff_r1__2_2 := -2365.895439+b1__1_2*r1__1_2+2*b1__2_2*r1__2_2+b1__2_3*r1__3_2-3.873365259*t13-65.50632175*t12+18.37256417*t11-1068.477762*r1__3_3-748.7401234*r1__3_2+666.8724958*r1__3_1-1303.778788*r1__2_3+2481.505981*r1__2_2-28.73393065*r1__2_1-210.0268079*r1__1_3-601.4685459*r1__1_2+319.0237491*r1__1_1;
diff_r1__3_2 := -91.67373642+b1__1_3*r1__1_2+b1__2_3*r1__2_2+2*b1__3_3*r1__3_2-94.90259936*t13-3.873365259*t12+38.92287358*t11-945.1329593*r1__3_3+2520.662892*r1__3_2-741.4079842*r1__3_1-1068.477762*r1__2_3-748.7401234*r1__2_2+666.8724957*r1__2_1+1087.842575*r1__1_3-981.8708336*r1__1_2-152.3287859*r1__1_1;
diff_r1__1_3 := 4702.012620+2*b1__1_1*r1__1_3+b1__1_2*r1__2_3+b1__1_3*r1__3_3-126.0393573*t13-115.7405279*t12+312.5409456*t11-9148.480288*r1__3_3+1087.842575*r1__3_2+2102.554696*r1__3_1-7724.631334*r1__2_3-210.0268079*r1__2_2+314.8874851*r1__2_1+19057.83604*r1__1_3+1514.592175*r1__1_2-3746.497095*r1__1_1;
diff_r1__2_3 := 6918.119675+b1__1_2*r1__1_3+2*b1__2_2*r1__2_3+b1__2_3*r1__3_3-98.27068398*t13+361.3041416*t12-115.7405279*t11-5656.776960*r1__3_3-1068.477762*r1__3_2+1815.507541*r1__3_1+25288.40206*r1__2_3-1303.778788*r1__2_2-5256.947355*r1__2_1-7724.631334*r1__1_3-210.0268075*r1__1_2+314.8874850*r1__1_1;
diff_r1__3_3 := -6160.643413+b1__1_3*r1__1_3+b1__2_3*r1__2_3+2*b1__3_3*r1__3_3+368.9870707*t13-98.27068398*t12-126.0393573*t11+25503.13466*r1__3_3-945.1329593*r1__3_2-2918.129645*r1__3_1-5656.776960*r1__2_3-1068.477762*r1__2_2+1815.507541*r1__2_1-9148.480288*r1__1_3+1087.842575*r1__1_2+2102.554697*r1__1_1;
g1:=r1^2+r4^2+r7^2-1;
g2:=r1*r2+r4*r5+r7*r8;
g3:=r1*r3+r4*r6+r7*r9;
g4:=r2^2+r5^2+r8^2-1;
g5:=r2*r3+r5*r6+r8*r9;
g6:=r3^2+r6^2+r9^2-1;
sum_sqr_distances:=(-34.5792590705286*r1+17.0635530183776*r4-2.30671047587914*r7+.533429751140582*t1-18.2777571201152+5.34368522421251*r2-2.6369060119417*r5+.356466130795291*r8-.0824332491501039*t2+31.8951297362284*r3-15.7390369840122*r6+2.12765778880161*r9-.492023587996135*t3)^2+(-63.880273658711*r2+31.5224925463221*r5-4.26132023642126*r8+.98543576110074*t2+5.20415366982094+5.34368522429473*r1-2.63690601164137*r4+.356466130730839*r7-.0824332491771907*t1+5.63520538121055*r3-2.78076015494726*r6+.375912834341308*r9-.0869303242748373*t3)^2+(-31.1892504431195*r3+15.3907123123387*r6-2.08057004842228*r9+.481134487803446*t3+16.460313265992+31.8951297380977*r1-15.7390369788429*r4+2.12765778852522*r7-.492023588007365*t1+5.63520538145412*r2-2.78076015435067*r5+.375912834360444*r8-.086930324248257*t2)^2+(-20.1477543311821*r1+7.55350991666248*r4+20.8920429846501*r7+.506802937949341*t1-13.5125935907312+7.82881106057577*r2-2.9350666574299*r5-8.1180192368411*r8-.19692837123503*t2+18.2686582928059*r3-6.84902591482382*r6-18.9435302828672*r9-.459535566231505*t3)^2+(-36.6286508786965*r2+13.7322935856179*r5+37.9817688885839*r8+.921368583909397*t2-34.3325081149272+7.82881105959312*r1-2.93506665641182*r4-8.11801923407865*r7-.196928371252735*t1+7.29448206284227*r3-2.7347436184352*r6-7.56395131161327*r9-.183487691946913*t3)^2+(-22.7328194787774*r3+8.52266582628087*r6+23.5726043680174*r9+.5718284781815*t3+29.2151845257836+18.2686582944502*r1-6.84902591329222*r4-18.9435302865599*r7-.459535566194367*t1+7.2944820644144*r2-2.73474361877224*r5-7.56395131566163*r8-.183487691915587*t2)^2+(-33.7370053449296*r1+38.9258336170637*r4-60.1132663033064*r7+.996042632734926*t1-4.98803179031169+1.08999592941726*r2-1.25763978603382*r5+1.94217639880877*r8-.0321807583045741*t2+1.8259306385537*r3-2.10676292990297*r6+3.25347948355056*r9-.0539083045951421*t3)^2+(-25.0073507347857*r2+28.8535382444551*r5-44.5585943124607*r8+.738310564652533*t2+4.93804814541239+1.08999592907818*r1-1.25763978578301*r4+1.94217639918338*r7-.0321807583046471*t1+14.8482131236429*r3-17.1319021236029*r6+26.4568411007238*r9-.438374809459876*t3)^2+(-8.9977349393474*r3+10.3816070715198*r6-16.0323428536159*r9+.265646802616713*t3-2.58161819993127+1.82593063830369*r1-2.1067629296129*r4+3.25347948341273*r7-.0539083045952243*t1+14.8482131262288*r2-17.1319021246606*r5+26.4568410945*r8-.438374809459549*t2)^2+(-56.6725654800191*r1-43.1813795970078*r4-58.3887310201578*r7+.980525398272198*t1+49.0916863573367+5.78296846451955*r2+4.40630407967054*r5+5.95808902028457*r8-.100054539766321*t2+5.50887455808305*r3+4.19745958942679*r6+5.67569496876523*r9-.0953122798368383*t3)^2+(-28.0870406673401*r2-21.400781038534*r5-28.937575861748*r8+.48595041543225*t2+19.4018945247715+5.78296846409823*r1+4.40630407900952*r4+5.95808902050728*r7-.100054539765808*t1+28.302910438969*r3+21.5652619383799*r6+29.1599826218892*r9-.489685305322089*t3)^2+(-30.836717169852*r3-23.4958833859599*r6-31.7705184032215*r9+.533524186319051*t3-30.3978529558191+5.50887455772671*r1+4.19745958933706*r4+5.67569496903033*r7-.0953122798304703*t1+28.3029104392002*r2+21.5652619411541*r5+29.1599826221612*r8-.489685305291881*t2)^2+(-57.8205382202822*r1-33.3924384768228*r4-34.894277849845*r7+.850839830317679*t1-12.5672620764049+10.7191476042258*r2+6.19050752309509*r5+6.46892827657879*r8-.157734223996209*t2+21.7070640341081*r3+12.5362340547815*r6+13.1000566054682*r9-.319423430626817*t3)^2+(-56.6217108878986*r2-32.7000933511969*r5-34.1707942023208*r8+.833198866184046*t2-1.62153141633832+10.7191476020505*r1+6.19050752434099*r4+6.46892827744991*r7-.15773422401843*t1+22.9548337765416*r3+13.2568443369017*r6+13.8530766468144*r9-.337784591359295*t3)^2+(-21.4717881919623*r3-12.4003578706288*r6-12.9580693345224*r9+.315961303486945*t3+6.66922254237138+21.7070640313443*r1+12.5362340557455*r4+13.1000566039334*r7-.319423430651098*t1+22.9548337782773*r2+13.2568443352531*r5+13.8530766433259*r8-.337784591337385*t2)^2+(-65.9377723150761*r1-12.678533244879*r4-65.0293030974106*r7+.955841723390729*t1+11.5835324790697+7.68982596884022*r2+1.47860188124137*r5+7.5838780437921*r8-.111472624096221*t2+11.9049207597142*r3+2.28908148248606*r6+11.7408986260292*r9-.17257513524552*t3)^2+(-49.571881779333*r2-9.53169524975009*r5-48.8888964859373*r8+.718599844159693*t2+14.0432100287845+7.6898259693682*r1+1.47860188123288*r4+7.58387804187672*r7-.111472624065637*t1+30.0526392395535*r3+5.77852985094098*r6+29.6385837317152*r9-.435646602436962*t3)^2+(-22.4583184276267*r3-4.31829173040524*r6-22.1488950108346*r9+.32555843242268*t3-12.0350031966363+11.904920759546*r1+2.28908148261245*r4+11.7408986234862*r7-.172575135204147*t1+30.0526392370654*r2+5.77852985129325*r5+29.6385837327811*r8-.435646602452044*t2)^2+(-40.2818933934206*r1+37.3415461911863*r4-17.5121421408269*r7+.688992135032061*t1+41.1079956272437+18.3480006638221*r2-17.0087018409058*r5+7.97660607993154*r8-.313829045441664*t2+19.8946971640516*r3-18.4424983689296*r6+8.64901659965967*r9-.340284150555745*t3)^2+(-39.9505168284587*r2+37.0343582140755*r5-17.3680795584234*r8+.68332418286889*t2-19.0229410779165+18.3480006624515*r1-17.0087018386759*r4+7.97660607742464*r7-.313829045473842*t1+20.0751637635565*r3-18.6097919416627*r6+8.72747261243189*r9-.343370898899277*t3)^2+(-36.6974975173012*r3+34.0187906543919*r6-15.9538626068613*r9+.627683682093741*t3-20.0272581058992+19.8946971612448*r1-18.4424983632605*r4+8.64901659883117*r7-.340284150571336*t1+20.0751637622239*r2-18.609791938382*r5+8.72747261433876*r8-.343370898879803*t2)^2+(-19.9003018252227*r1+2.10723968998899*r4-5.99083386660634*r7+.268355364500816*t1-6.36217996635187+32.4452671740533*r2-3.43562400998522*r5+9.76739986562233*r8-.437524092703596*t2+5.19806634264006*r3-.550422390906632*r6+1.56483817002714*r9-.0700958709468267*t3)^2+(-54.7542305404035*r2+5.79791647536792*r5-16.4833428943737*r8+.738360233327084*t2+12.3293109763825+32.4452671726755*r1-3.43562401018043*r4+9.76739986673756*r7-.437524092757061*t1+3.1084479408544*r3-.329153041695288*r6+.935774510511675*r9-.0419173883795401*t3)^2+(-73.6585215390464*r3+7.79968874305612*r6-22.1743353083965*r9+.99328440217389*t3-10.5500615738974+5.19806634229016*r1-.550422390946524*r4+1.5648381697327*r7-.0700958709446797*t1+3.10844794077716*r2-.329153041700441*r5+.935774510228754*r8-.041917388373134*t2)^2;
h:=sum_sqr_distances+g1*b1+g2*b2+g3*b3+g4*b4+g5*b5+g6*b6;
diff_t1:=diff(h,t1);
diff_t2:=diff(h,t2);
diff_t3:=diff(h,t3);
diff_r1:=diff(h,r1);
diff_r2:=diff(h,r2);
diff_r3:=diff(h,r3);
diff_r4:=diff(h,r4);
diff_r5:=diff(h,r5);
diff_r6:=diff(h,r6);
diff_r7:=diff(h,r7);
diff_r8:=diff(h,r8);
diff_r9:=diff(h,r9);
diff_b1:=diff(h,b1);
diff_b2:=diff(h,b2);
diff_b3:=diff(h,b3);
diff_b4:=diff(h,b4);
diff_b5:=diff(h,b5);
diff_b6:=diff(h,b6);
vars := [op(indets(h, And(name, Non(constant))))];
polysys:={g1_1_1,g1_1_2,g1_1_3,g1_2_2,g1_2_3,g1_3_3,diff_t11,diff_t12,diff_t13,diff_r1__1_1,diff_r1__1_2,diff_r1__1_3,diff_r1__2_1,diff_r1__2_2,diff_r1__2_3,diff_r1__3_1,diff_r1__3_2,diff_r1__3_3};
polysys:={diff_t1,diff_t2,diff_t3,diff_r1,diff_r4,diff_r7,diff_r2,diff_r5,diff_r8,diff_r3,diff_r6,diff_r9,diff_b1,diff_b2,diff_b3,diff_b4,diff_b5,diff_b6};
sols := CodeTools:-Usage(RootFinding:-Isolate(polysys, vars, output = numeric, method = RS));
P.<b1__1_1, b1__1_2, b1__1_3, b1__2_2, b1__2_3, b1__3_3, r1__1_1, r1__1_2, r1__1_3, r1__2_1, r1__2_2, r1__2_3, r1__3_1, r1__3_2, r1__3_3, t11, t12, t13>=PolynomialRing(QQ,order='degrevlex')
g1_1_1 = r1__1_1^2+r1__1_2^2+r1__1_3^2-1
g1_1_2 = r1__1_1*r1__2_1+r1__1_2*r1__2_2+r1__1_3*r1__2_3
g1_1_3 = r1__1_1*r1__3_1+r1__1_2*r1__3_2+r1__1_3*r1__3_3
g1_2_2 = r1__2_1^2+r1__2_2^2+r1__2_3^2-1
g1_2_3 = r1__2_1*r1__3_1+r1__2_2*r1__3_2+r1__2_3*r1__3_3
g1_3_3 = r1__3_1^2+r1__3_2^2+r1__3_3^2-1
diff_t11 = 74597*r1__2_1*(1/5996)-32999*t12*(1/8859)-47724*t13*(1/13471)-285739/3355+51151*t11*(1/4872)-227375*r1__3_3*(1/1804)+250313*r1__3_2*(1/6431)+21041*r1__3_1*(1/506)-946989*r1__2_3*(1/8182)+95225*r1__2_2*(1/5183)+270973*r1__1_3*(1/867)-201713*r1__1_2*(1/6275)-333665*r1__1_1*(1/6466)
diff_t12 = -132813*r1__2_1*(1/1250)+86862*t12*(1/9377)-49495*t13*(1/14984)-32999*t11*(1/8859)-205484*r1__3_3*(1/2091)-101885*r1__3_2*(1/26304)+77695*r1__3_1*(1/2206)+305302*r1__2_3*(1/845)-336768*r1__2_2*(1/5141)-946989*r1__1_3*(1/8182)+95225*r1__1_2*(1/5183)+74597*r1__1_1*(1/5996)+163365/1132
diff_t13 = 77695*r1__2_1*(1/2206)-49495*t12*(1/14984)+17194*t13*(1/1405)-47724*t11*(1/13471)+827638*r1__3_3*(1/2243)-303024*r1__3_2*(1/3193)-189788*r1__3_1*(1/4609)-205484*r1__2_3*(1/2091)-71971*r1__2_2*(1/18581)-227375*r1__1_3*(1/1804)+338629*r1__1_2*(1/8700)+21041*r1__1_1*(1/506)+43984/1749
diff_r1__1_1 = -(754990/629)*r1__2_1+(74597/5996)*t12+(21041/506)*t13-(333665/6466)*t11+(1364558/649)*r1__3_3-(111657/733)*r1__3_2-(1224599/1174)*r1__3_1+(249076/791)*r1__2_3+(376129/1179)*r1__2_2-(1288795/344)*r1__1_3-(936443/568)*r1__1_2+(1265223/371)*r1__1_1+1084247/1835+2*b1__1_1*r1__1_1+b1__1_2*r1__2_1+b1__1_3*r1__3_1
diff_r1__2_1 = (2227310/811)*r1__2_1-(132813/1250)*t12+(77695/2206)*t13-1929092/505+(74597/5996)*t11+b1__1_2*r1__1_1+2*b1__2_2*r1__2_1+b1__2_3*r1__3_1+(2046077/1127)*r1__3_3+(366113/549)*r1__3_2-(1509452/1137)*r1__3_1-(20570435/3913)*r1__2_3-(240503/8370)*r1__2_2+(772419/2453)*r1__1_3+(376129/1179)*r1__1_2-(754990/629)*r1__1_1
diff_r1__3_1 = b1__1_3*r1__1_1+b1__2_3*r1__2_1+2*b1__3_3*r1__3_1-(1509452/1137)*r1__2_1+(77695/2206)*t12-(189788/4609)*t13+3284794/809+(21041/506)*t11-(3218697/1103)*r1__3_3-(761426/1027)*r1__3_2+(316876/93)*r1__3_1+(2046077/1127)*r1__2_3+(366113/549)*r1__2_2+(1364558/649)*r1__1_3-(111657/733)*r1__1_2-(1224599/1174)*r1__1_1
diff_r1__1_2 = (376129/1179)*r1__2_1+(95225/5183)*t12+(338629/8700)*t13-614841/889+2*b1__1_1*r1__1_2+b1__1_2*r1__2_2+b1__1_3*r1__3_2-(201713/6275)*t11+(608104/559)*r1__3_3-(235649/240)*r1__3_2-(111657/733)*r1__3_1-(258543/1231)*r1__2_3-(1749672/2909)*r1__2_2+(2012893/1329)*r1__1_3+(3640570/1581)*r1__1_2-(936443/568)*r1__1_1
diff_r1__2_2 = b1__1_2*r1__1_2+2*b1__2_2*r1__2_2+b1__2_3*r1__3_2-(240503/8370)*r1__2_1-(336768/5141)*t12-(71971/18581)*t13-1244461/526+(95225/5183)*t11-(744729/697)*r1__3_3-(852815/1139)*r1__3_2+(366113/549)*r1__3_1-(430247/330)*r1__2_3+(1451681/585)*r1__2_2-(336883/1604)*r1__1_3-(1749672/2909)*r1__1_2+(376129/1179)*r1__1_1
diff_r1__3_2 = (366113/549)*r1__2_1-(101885/26304)*t12-(303024/3193)*t13+b1__1_3*r1__1_2+b1__2_3*r1__2_2+2*b1__3_3*r1__3_2+(250313/6431)*t11-(1350595/1429)*r1__3_3+(889794/353)*r1__3_2-(761426/1027)*r1__3_1-(744729/697)*r1__2_3-(852815/1139)*r1__2_2+(608104/559)*r1__1_3-(235649/240)*r1__1_2-(111657/733)*r1__1_1-203149/2216
diff_r1__1_3 = (772419/2453)*r1__2_1-(946989/8182)*t12-(227375/1804)*t13+(270973/867)*t11-(2552426/279)*r1__3_3+(608104/559)*r1__3_2+(1364558/649)*r1__3_1-(1676245/217)*r1__2_3-(336883/1604)*r1__2_2+(1162528/61)*r1__1_3+(2012893/1329)*r1__1_2-(1288795/344)*r1__1_1+1490538/317+2*b1__1_1*r1__1_3+b1__1_2*r1__2_3+b1__1_3*r1__3_3
diff_r1__2_3 = -(20570435/3913)*r1__2_1+(305302/845)*t12-(205484/2091)*t13-(946989/8182)*t11-(2307965/408)*r1__3_3-(744729/697)*r1__3_2+(2046077/1127)*r1__3_1+(2452975/97)*r1__2_3-(430247/330)*r1__2_2-(1676245/217)*r1__1_3-(258543/1231)*r1__1_2+(249076/791)*r1__1_1+2601213/376+b1__1_2*r1__1_3+2*b1__2_2*r1__2_3+b1__2_3*r1__3_3
diff_r1__3_3 = (2046077/1127)*r1__2_1-(205484/2091)*t12+(827638/2243)*t13-794723/129-(227375/1804)*t11+b1__1_3*r1__1_3+b1__2_3*r1__2_3+2*b1__3_3*r1__3_3+(7574431/297)*r1__3_3-(1350595/1429)*r1__3_2-(3218697/1103)*r1__3_1-(2307965/408)*r1__2_3-(744729/697)*r1__2_2-(2552426/279)*r1__1_3+(608104/559)*r1__1_2+(1364558/649)*r1__1_1
I=Ideal(g1_1_1, g1_1_2, g1_1_3, g1_2_2, g1_2_3, g1_3_3, diff_t11, diff_t12, diff_t13, diff_r1__1_1, diff_r1__2_1, diff_r1__3_1, diff_r1__1_2, diff_r1__2_2, diff_r1__3_2, diff_r1__1_3, diff_r1__2_3, diff_r1__3_3)
P.<r1,r2,r3,r4,r5,r6,r7,r8,r9,t1,t2,t3,b1,b2,b3,b4,b5,b6>=PolynomialRing(QQ,order='degrevlex')
g1=r1^2+r4^2+r7^2-1
g2=r1*r2+r4*r5+r7*r8
g3=r1*r3+r4*r6+r7*r9
g4=r2^2+r5^2+r8^2-1
g5=r2*r3+r5*r6+r8*r9
g6=r3^2+r6^2+r9^2-1
sum_sqr_distances=(-34.5792590705286*r1+17.0635530183776*r4-2.30671047587914*r7+.533429751140582*t1-18.2777571201152+5.34368522421251*r2-2.6369060119417*r5+.356466130795291*r8-.0824332491501039*t2+31.8951297362284*r3-15.7390369840122*r6+2.12765778880161*r9-.492023587996135*t3)^2+(-63.880273658711*r2+31.5224925463221*r5-4.26132023642126*r8+.98543576110074*t2+5.20415366982094+5.34368522429473*r1-2.63690601164137*r4+.356466130730839*r7-.0824332491771907*t1+5.63520538121055*r3-2.78076015494726*r6+.375912834341308*r9-.0869303242748373*t3)^2+(-31.1892504431195*r3+15.3907123123387*r6-2.08057004842228*r9+.481134487803446*t3+16.460313265992+31.8951297380977*r1-15.7390369788429*r4+2.12765778852522*r7-.492023588007365*t1+5.63520538145412*r2-2.78076015435067*r5+.375912834360444*r8-.086930324248257*t2)^2+(-20.1477543311821*r1+7.55350991666248*r4+20.8920429846501*r7+.506802937949341*t1-13.5125935907312+7.82881106057577*r2-2.9350666574299*r5-8.1180192368411*r8-.19692837123503*t2+18.2686582928059*r3-6.84902591482382*r6-18.9435302828672*r9-.459535566231505*t3)^2+(-36.6286508786965*r2+13.7322935856179*r5+37.9817688885839*r8+.921368583909397*t2-34.3325081149272+7.82881105959312*r1-2.93506665641182*r4-8.11801923407865*r7-.196928371252735*t1+7.29448206284227*r3-2.7347436184352*r6-7.56395131161327*r9-.183487691946913*t3)^2+(-22.7328194787774*r3+8.52266582628087*r6+23.5726043680174*r9+.5718284781815*t3+29.2151845257836+18.2686582944502*r1-6.84902591329222*r4-18.9435302865599*r7-.459535566194367*t1+7.2944820644144*r2-2.73474361877224*r5-7.56395131566163*r8-.183487691915587*t2)^2+(-33.7370053449296*r1+38.9258336170637*r4-60.1132663033064*r7+.996042632734926*t1-4.98803179031169+1.08999592941726*r2-1.25763978603382*r5+1.94217639880877*r8-.0321807583045741*t2+1.8259306385537*r3-2.10676292990297*r6+3.25347948355056*r9-.0539083045951421*t3)^2+(-25.0073507347857*r2+28.8535382444551*r5-44.5585943124607*r8+.738310564652533*t2+4.93804814541239+1.08999592907818*r1-1.25763978578301*r4+1.94217639918338*r7-.0321807583046471*t1+14.8482131236429*r3-17.1319021236029*r6+26.4568411007238*r9-.438374809459876*t3)^2+(-8.9977349393474*r3+10.3816070715198*r6-16.0323428536159*r9+.265646802616713*t3-2.58161819993127+1.82593063830369*r1-2.1067629296129*r4+3.25347948341273*r7-.0539083045952243*t1+14.8482131262288*r2-17.1319021246606*r5+26.4568410945*r8-.438374809459549*t2)^2+(-56.6725654800191*r1-43.1813795970078*r4-58.3887310201578*r7+.980525398272198*t1+49.0916863573367+5.78296846451955*r2+4.40630407967054*r5+5.95808902028457*r8-.100054539766321*t2+5.50887455808305*r3+4.19745958942679*r6+5.67569496876523*r9-.0953122798368383*t3)^2+(-28.0870406673401*r2-21.400781038534*r5-28.937575861748*r8+.48595041543225*t2+19.4018945247715+5.78296846409823*r1+4.40630407900952*r4+5.95808902050728*r7-.100054539765808*t1+28.302910438969*r3+21.5652619383799*r6+29.1599826218892*r9-.489685305322089*t3)^2+(-30.836717169852*r3-23.4958833859599*r6-31.7705184032215*r9+.533524186319051*t3-30.3978529558191+5.50887455772671*r1+4.19745958933706*r4+5.67569496903033*r7-.0953122798304703*t1+28.3029104392002*r2+21.5652619411541*r5+29.1599826221612*r8-.489685305291881*t2)^2+(-57.8205382202822*r1-33.3924384768228*r4-34.894277849845*r7+.850839830317679*t1-12.5672620764049+10.7191476042258*r2+6.19050752309509*r5+6.46892827657879*r8-.157734223996209*t2+21.7070640341081*r3+12.5362340547815*r6+13.1000566054682*r9-.319423430626817*t3)^2+(-56.6217108878986*r2-32.7000933511969*r5-34.1707942023208*r8+.833198866184046*t2-1.62153141633832+10.7191476020505*r1+6.19050752434099*r4+6.46892827744991*r7-.15773422401843*t1+22.9548337765416*r3+13.2568443369017*r6+13.8530766468144*r9-.337784591359295*t3)^2+(-21.4717881919623*r3-12.4003578706288*r6-12.9580693345224*r9+.315961303486945*t3+6.66922254237138+21.7070640313443*r1+12.5362340557455*r4+13.1000566039334*r7-.319423430651098*t1+22.9548337782773*r2+13.2568443352531*r5+13.8530766433259*r8-.337784591337385*t2)^2+(-65.9377723150761*r1-12.678533244879*r4-65.0293030974106*r7+.955841723390729*t1+11.5835324790697+7.68982596884022*r2+1.47860188124137*r5+7.5838780437921*r8-.111472624096221*t2+11.9049207597142*r3+2.28908148248606*r6+11.7408986260292*r9-.17257513524552*t3)^2+(-49.571881779333*r2-9.53169524975009*r5-48.8888964859373*r8+.718599844159693*t2+14.0432100287845+7.6898259693682*r1+1.47860188123288*r4+7.58387804187672*r7-.111472624065637*t1+30.0526392395535*r3+5.77852985094098*r6+29.6385837317152*r9-.435646602436962*t3)^2+(-22.4583184276267*r3-4.31829173040524*r6-22.1488950108346*r9+.32555843242268*t3-12.0350031966363+11.904920759546*r1+2.28908148261245*r4+11.7408986234862*r7-.172575135204147*t1+30.0526392370654*r2+5.77852985129325*r5+29.6385837327811*r8-.435646602452044*t2)^2+(-40.2818933934206*r1+37.3415461911863*r4-17.5121421408269*r7+.688992135032061*t1+41.1079956272437+18.3480006638221*r2-17.0087018409058*r5+7.97660607993154*r8-.313829045441664*t2+19.8946971640516*r3-18.4424983689296*r6+8.64901659965967*r9-.340284150555745*t3)^2+(-39.9505168284587*r2+37.0343582140755*r5-17.3680795584234*r8+.68332418286889*t2-19.0229410779165+18.3480006624515*r1-17.0087018386759*r4+7.97660607742464*r7-.313829045473842*t1+20.0751637635565*r3-18.6097919416627*r6+8.72747261243189*r9-.343370898899277*t3)^2+(-36.6974975173012*r3+34.0187906543919*r6-15.9538626068613*r9+.627683682093741*t3-20.0272581058992+19.8946971612448*r1-18.4424983632605*r4+8.64901659883117*r7-.340284150571336*t1+20.0751637622239*r2-18.609791938382*r5+8.72747261433876*r8-.343370898879803*t2)^2+(-19.9003018252227*r1+2.10723968998899*r4-5.99083386660634*r7+.268355364500816*t1-6.36217996635187+32.4452671740533*r2-3.43562400998522*r5+9.76739986562233*r8-.437524092703596*t2+5.19806634264006*r3-.550422390906632*r6+1.56483817002714*r9-.0700958709468267*t3)^2+(-54.7542305404035*r2+5.79791647536792*r5-16.4833428943737*r8+.738360233327084*t2+12.3293109763825+32.4452671726755*r1-3.43562401018043*r4+9.76739986673756*r7-.437524092757061*t1+3.1084479408544*r3-.329153041695288*r6+.935774510511675*r9-.0419173883795401*t3)^2+(-73.6585215390464*r3+7.79968874305612*r6-22.1743353083965*r9+.99328440217389*t3-10.5500615738974+5.19806634229016*r1-.550422390946524*r4+1.5648381697327*r7-.0700958709446797*t1+3.10844794077716*r2-.329153041700441*r5+.935774510228754*r8-.041917388373134*t2)^2
h=sum_sqr_distances+g1*b1+g2*b2+g3*b3+g4*b4+g5*b5+g6*b6
diff_t1=diff(h,t1)
diff_t2=diff(h,t2)
diff_t3=diff(h,t3)
diff_r1=diff(h,r1)
diff_r2=diff(h,r2)
diff_r3=diff(h,r3)
diff_r4=diff(h,r4)
diff_r5=diff(h,r5)
diff_r6=diff(h,r6)
diff_r7=diff(h,r7)
diff_r8=diff(h,r8)
diff_r9=diff(h,r9)
diff_b1=diff(h,b1)
diff_b2=diff(h,b2)
diff_b3=diff(h,b3)
diff_b4=diff(h,b4)
diff_b5=diff(h,b5)
diff_b6=diff(h,b6)
I=Ideal(g1,g2,g3,g4,g5,g6,diff_t1,diff_t2,diff_t3,diff_r1,diff_r2,diff_r3,diff_r4,diff_r5,diff_r6,diff_r7,diff_r8,diff_r9)
I.groebner_basis()
I.variety(RR)
12 22 solutions the Maple found (all the real solutions that solve the equations):
The goal is to get the same solutions from sage.
[[b1__1_1 = -899.5942504, b1__1_2 = 3037.238105, b1__1_3 = -3600.559806, b1__2_2 = -1064.022119, b1__2_3 = 889.3168953, b1__3_3 = -2555.002632, r1__1_1 = .7481491832, r1__1_2 = .6388182437, r1__1_3 = .1793991396, r1__2_1 = -.6472289487, r1__2_2 = .7621522200, r1__2_3 = -0.1478788196e-1, r1__3_1 = -.1461762213, r1__3_2 = -.1050487747, r1__3_3 = .9836652211, t11 = .1547590859, t12 = -20.00614350, t13 = -39.11689082],
[b1__1_1 = -2536.737834, b1__1_2 = -384.6438623, b1__1_3 = 2968.302582, b1__2_2 = 206.1653350, b1__2_3 = -4119.641687, b1__3_3 = -2528.112061, r1__1_1 = .3433731644, r1__1_2 = -.9364692403, r1__1_3 = -0.7155579556e-1, r1__2_1 = .8767823107, r1__2_2 = .2923108835, r1__2_3 = .3818469942, r1__3_1 = -.3366714267, r1__3_2 = -.1938548665, r1__3_3 = .9214513775, t11 = 8.655455565, t12 = -14.90305773, t13 = -32.28108776],
[b1__1_1 = -2672.661895, b1__1_2 = 5518.566630, b1__1_3 = 194.0920868, b1__2_2 = -1264.096636, b1__2_3 = -39.30164061, b1__3_3 = -3167.161180, r1__1_1 = .6407595964, r1__1_2 = .1199696331, r1__1_3 = -.7583102444, r1__2_1 = -.1297813989, r1__2_2 = .9904267144, r1__2_3 = 0.4702884205e-1, r1__3_1 = .7566927568, r1__3_2 = 0.6828038249e-1, r1__3_3 = .6501952485, t11 = 21.88635414, t12 = -19.14986568, t13 = -26.72096477],
[b1__1_1 = -2415.819809, b1__1_2 = -247.8725031, b1__1_3 = -81.80497732, b1__2_2 = 578.5002146, b1__2_3 = -3308.869199, b1__3_3 = -632.2913675, r1__1_1 = .5855751733, r1__1_2 = -.6602102503, r1__1_3 = .4703447053, r1__2_1 = .5367090123, r1__2_2 = .7506041613, r1__2_3 = .3854047601, r1__3_1 = -.6074908662, r1__3_2 = 0.2675478306e-1, r1__3_3 = .7938759532, t11 = -4.150377642, t12 = -12.85905997, t13 = -25.75853354],
[b1__1_1 = 305.8128664, b1__1_2 = 1219.200904, b1__1_3 = -4379.467945, b1__2_2 = -1232.280299, b1__2_3 = 1920.610983, b1__3_3 = -1454.259739, r1__1_1 = .4050262785, r1__1_2 = .2045666996, r1__1_3 = -.8911263542, r1__2_1 = -.7397436412, r1__2_2 = -.4994839335, r1__2_3 = -.4508826294, r1__3_1 = .5373388680, r1__3_2 = -.8418243674, r1__3_3 = 0.5097720352e-1, t11 = 19.84743424, t12 = -23.75250545, t13 = -21.83242508],
[b1__1_1 = -2561.237930, b1__1_2 = 5363.493736, b1__1_3 = -370.0386489, b1__2_2 = -1179.131772, b1__2_3 = 977.8467630, b1__3_3 = -2792.263176, r1__1_1 = .4494018895, r1__1_2 = 0.2797468228e-1, r1__1_3 = -.8928915717, r1__2_1 = -.2109820307, r1__2_2 = .9745577061, r1__2_3 = -0.7565619745e-1, r1__3_1 = .8680579038, r1__3_2 = .2223841151, r1__3_3 = .4438702298, t11 = 24.07022969, t12 = -15.38154019, t13 = -18.48056832],
[b1__1_1 = 549.2162023, b1__1_2 = 87.77137600, b1__1_3 = -1198.811469, b1__2_2 = -495.0197371, b1__2_3 = -1765.778607, b1__3_3 = 676.3321754, r1__1_1 = -.3382824498, r1__1_2 = .1447284170, r1__1_3 = -.9298487347, r1__2_1 = .8458105804, r1__2_2 = -.3863831243, r1__2_3 = -.3678485331, r1__3_1 = -.4125159951, r1__3_2 = -.9109126010, r1__3_3 = 0.8293803447e-2, t11 = 30.93253221, t12 = 2.460529285, t13 = -15.53626680],
[b1__1_1 = -1223.590670, b1__1_2 = 1841.223573, b1__1_3 = 4119.421790, b1__2_2 = 671.1576084, b1__2_3 = -1371.054802, b1__3_3 = -903.8038187, r1__1_1 = -.4128628600, r1__1_2 = -0.7780371756e-1, r1__1_3 = -.9074639609, r1__2_1 = .2440176361, r1__2_2 = .9504710646, r1__2_3 = -.1925101259, r1__3_1 = -.8774962405, r1__3_2 = .3009174918, r1__3_3 = .3734287229, t11 = 34.69684826, t12 = 8.155443592, t13 = -11.33088510],
[b1__1_1 = -6327.273294, b1__1_2 = 1759.073834, b1__1_3 = 11207.90070, b1__2_2 = -1890.858242, b1__2_3 = 1891.950907, b1__3_3 = -9760.176346, r1__1_1 = .3766394529, r1__1_2 = -.6180780439, r1__1_3 = .6900161260, r1__2_1 = -.7796490396, r1__2_2 = -.6137728279, r1__2_3 = -.1242187218, r1__3_1 = .5002900135, r1__3_2 = -.4911847385, r1__3_3 = -.7130550154, t11 = -29.31512316, t12 = -33.00946996, t13 = 8.767454469],
[b1__1_1 = -3304.385960, b1__1_2 = -4044.369886, b1__1_3 = 10477.87582, b1__2_2 = -1108.617283, b1__2_3 = 5826.342875, b1__3_3 = -9844.297818, r1__1_1 = -.5741762462, r1__1_2 = -.7368503356, r1__1_3 = .3568938514, r1__2_1 = .7828801205, r1__2_2 = -.3665428545, r1__2_3 = .5027375585, r1__3_1 = .2396254476, r1__3_2 = -.5680650656, r1__3_3 = -.7873256798, t11 = -18.36435913, t12 = -34.30593158, t13 = 13.14730960],
[b1__1_1 = -6612.907618, b1__1_2 = 2424.756675, b1__1_3 = 12384.95493, b1__2_2 = -1691.366479, b1__2_3 = 2578.504868, b1__3_3 = -9210.975576, r1__1_1 = .5157887216, r1__1_2 = .4217561249, r1__1_3 = .7457102425, r1__2_1 = -.4865104064, r1__2_2 = .8606564397, r1__2_3 = -.1502601654, r1__3_1 = .7051734674, r1__3_2 = .2852932946, r1__3_3 = -.6491056285, t11 = -24.37008827, t12 = -14.94325716, t13 = 16.06559316],
[b1__1_1 = -4846.263193, b1__1_2 = -3105.528126, b1__1_3 = 12948.69518, b1__2_2 = -279.0100568, b1__2_3 = 4363.645053, b1__3_3 = -9431.494358, r1__1_1 = -.8785017929, r1__1_2 = .3491284937, r1__1_3 = .3261041166, r1__2_1 = .4347305027, r1__2_2 = .8672555961, r1__2_3 = .2426460819, r1__3_1 = -.1981009589, r1__3_2 = .3549324245, r1__3_3 = -.9136624016, t11 = -11.20139735, t12 = -13.01981734, t13 = 27.02745538]]
[[b1 = -6405.651236, b2 = 1097.011411, b3 = 6432.603484, b4 = -3486.767199, b5 = 2786.505511, b6 = -2066.461540, r1 = -.2735272365, r2 = -.4584555697, r3 = -.8455775195, r4 = .9594115168, r5 = -0.6729917007e-1, r6 = -.2738619418, r7 = -0.6864686727e-1, r8 = .8861655107, r9 = -.4582557095, t1 = -31.53181188, t2 = -3.035192389, t3 = -64.33835896],
[b1 = -663.0587184, b2 = 1206.645590, b3 = -935.8274112, b4 = -2827.613858, b5 = 1260.996624, b6 = -676.9562322, r1 = .8750945364, r2 = -.1796187040, r3 = -.4493847723, r4 = -.4474525861, r5 = -.6540696330, r6 = -.6099008922, r7 = -.1843793253, r8 = .7347993170, r9 = -.6527436159, t1 = 24.66279957, t2 = 6.035593829, t3 = -55.14919482],
[b1 = -5334.007938, b2 = -147.4788994, b3 = 6381.507728, b4 = -1654.278431, b5 = 3038.269718, b6 = -2677.888758, r1 = 0.1140465611e-1, r2 = -.8980265159, r3 = -.4397934863, r4 = .8207951804, r5 = .2596099997, r6 = -.5088201253, r7 = -.5711087512, r8 = .3551774554, r9 = -.7400565989, t1 = -38.50311082, t2 = -47.94244185, t3 = -52.00642291],
[b1 = -3711.964527, b2 = 3878.643381, b3 = -871.0350173, b4 = -4720.758037, b5 = 5739.216423, b6 = -3328.445679, r1 = -.3452025304, r2 = -.5900255619, r3 = -.7298664599, r4 = -.5720856277, r5 = .7487785844, r6 = -.3347367116, r7 = .7440115910, r8 = .3019941520, r9 = -.5960254060, t1 = 4.420063381, t2 = -25.83338983, t3 = -49.39760950],
[b1 = -6771.196603, b2 = 3126.930183, b3 = 3836.296676, b4 = -2119.174006, b5 = -1053.266216, b6 = -647.0603393, r1 = -0.5288779601e-1, r2 = .2948075346, r3 = -.9540919235, r4 = .9579998211, r5 = -.2546859938, r6 = -.1318005592, r7 = .2818496477, r8 = .9209905331, r9 = .2689557845, t1 = -1.568329794, t2 = 41.81946440, t3 = -47.91552928],
[b1 = -426.4877331, b2 = 1777.234977, b3 = -1396.594665, b4 = -4704.976941, b5 = 6548.202362, b6 = -2797.527674, r1 = .6772160397, r2 = -.2551001296, r3 = -.6901466217, r4 = -.3808728417, r5 = .6809888166, r6 = -.6254519247, r7 = .6295349982, r8 = .6864241804, r9 = .3640158382, t1 = 51.90734298, t2 = -10.61796786, t3 = -36.65155778],
[b1 = -8345.957097, b2 = -5992.141964, b3 = 9771.580880, b4 = -9281.942970, b5 = 14067.10170, b6 = -7238.069438, r1 = -.4112595863, r2 = -.5902847582, r3 = -.6945714196, r4 = -.9059223617, r5 = .1803855392, r6 = .3831001589, r7 = .1008475446, r8 = -.7867813937, r9 = .6089374444, t1 = -25.97531493, t2 = -66.23689116, t3 = -31.46159720],
[b1 = -5343.147703, b2 = 183.2763579, b3 = 5426.240402, b4 = -11788.39198, b5 = 16609.81042, b6 = -7181.846374, r1 = -.2988639430, r2 = -.6827291572, r3 = -.6667542587, r4 = .7307518828, r5 = -.6130817579, r6 = .3002206588, r7 = -.6137442703, r8 = -.3975068000, r9 = .6821336486, t1 = -51.64655336, t2 = -55.45375783, t3 = -29.08482105],
[b1 = -4075.023103, b2 = 6883.346857, b3 = -1945.068213, b4 = -3311.752332, b5 = 3039.918275, b6 = -921.4062369, r1 = .5825663050, r2 = .8122102291, r3 = 0.3051301418e-1, r4 = -.8079724338, r5 = .5827884450, r6 = -0.8682266235e-1, r7 = 0.8830088656e-1, r8 = -0.2592628327e-1, r9 = -.9957563865, t1 = 23.77615250, t2 = 40.91304243, t3 = -28.03945694],
[b1 = -6187.757714, b2 = 838.7708899, b3 = 6066.514065, b4 = -441.1250637, b5 = -851.5589307, b6 = -1128.553632, r1 = .2065219306, r2 = .9631666887, r3 = -.1722167877, r4 = .8113374284, r5 = -.2669555392, r6 = -.5200637629, r7 = -.5468823178, r8 = -0.3232135326e-1, r9 = -.8365853576, t1 = -21.72396020, t2 = 62.90274148, t3 = -22.56105906],
[b1 = -3971.638517, b2 = 5211.591913, b3 = -2909.389068, b4 = -3415.632999, b5 = 2348.040874, b6 = -1624.438321, r1 = -.6376325216, r2 = -.7577247398, r3 = -.1388451874, r4 = -.6573617358, r5 = .6291742407, r6 = -.4147473004, r7 = .4016221056, r8 = -.1731848536, r9 = -.8992812078, t1 = -24.08083201, t2 = -42.77528618, t3 = -16.88667732],
[b1 = -1107.397052, b2 = -1352.354473, b3 = 559.8732630, b4 = -682.7595891, b5 = 1919.821403, b6 = -459.8284277, r1 = -.5540008534, r2 = .8147846552, r3 = -.1709064659, r4 = .3838720628, r5 = .4321713781, r6 = .8160086638, r7 = .7387322206, r8 = .3864632785, r9 = -.5521963786, t1 = -3.462811975, t2 = 66.31134317, t3 = -11.39008932],
[b1 = -1858.210974, b2 = -1443.942073, b3 = -117.7029385, b4 = -541.5565807, b5 = 635.7869277, b6 = -1292.459322, r1 = -.6056473654, r2 = .7430910328, r3 = -.2846172619, r4 = -.4570833521, r5 = -.6176622647, r6 = -.6399751058, r7 = .6513571049, r8 = .2575054246, r9 = -.7137400635, t1 = -8.170594439, t2 = 73.18220544, t3 = -5.574473809],
[b1 = -2879.955409, b2 = -7247.457464, b3 = 7603.360417, b4 = -10386.60081, b5 = 15680.46439, b6 = -7326.146084, r1 = .8898800024, r2 = -.3777475308, r3 = .2557740885, r4 = -.2159488982, r5 = .1450669474, r6 = .9655680474, r7 = -.4018453119, r8 = -.9144738289, r9 = 0.4751801242e-1, t1 = 25.17284844, t2 = -56.81452504, t3 = 4.483386705],
[b1 = -2594.500775, b2 = -8451.746216, b3 = 8378.142403, b4 = -8445.314669, b5 = 16539.21636, b6 = -7726.257147, r1 = -.8887915121, r2 = .4283658935, r3 = .1629487933, r4 = .2014992422, r5 = 0.4589445400e-1, r6 = .9784128753, r7 = -.4116402597, r8 = -.9024391172, r9 = .1271060041, t1 = -72.41708628, t2 = -1.180372243, t3 = 11.62696253],
[b1 = -3386.721976, b2 = -9232.755155, b3 = 8735.844228, b4 = -8449.692294, b5 = 14675.00399, b6 = -7470.241638, r1 = .7020126763, r2 = -.7115667558, r3 = -0.2917112402e-1, r4 = -.6916841933, r5 = -.6714995329, r6 = -.2658220344, r7 = -.1695617265, r8 = -.2067876432, r9 = .9635806617, t1 = 22.56779868, t2 = -57.27997094, t3 = 12.66120812],
[b1 = -2344.035915, b2 = 3947.708934, b3 = -879.5603548, b4 = -4851.200663, b5 = 7270.443175, b6 = -2742.299953, r1 = .4417879292, r2 = .7119951524, r3 = .5457896376, r4 = -.5754568251, r5 = .6916351055, r6 = -.4364519713, r7 = .6882389614, r8 = .1212591594, r9 = -.7152785110, t1 = 49.50818048, t2 = 50.77308450, t3 = 32.05737927],
[b1 = -4793.592445, b2 = -9508.057512, b3 = 9867.840732, b4 = -7363.405470, b5 = 13671.66420, b6 = -8641.244153, r1 = -.8607833310, r2 = .4979577238, r3 = .1053098401, r4 = -.5071531704, r5 = -.8216734210, r6 = -.2600931582, r7 = -0.4298510042e-1, r8 = -.2772920743, r9 = .9598236227, t1 = -55.37291300, t2 = 22.92886207, t3 = 35.86152727],
[b1 = -1143.596369, b2 = 1527.374908, b3 = 1286.733075, b4 = -1246.223196, b5 = 880.0474364, b6 = -176.7812153, r1 = .5096005566, r2 = -.5301666308, r3 = .6776655638, r4 = -0.1872847237e-1, r5 = .7805852008, r6 = .6247687481, r7 = .8602072524, r8 = .3310741426, r9 = -.3878574417, t1 = 62.72704338, t2 = -19.19220727, t3 = 40.08281887],
[b1 = -6277.680746, b2 = -547.8928701, b3 = 6669.122409, b4 = -11168.08424, b5 = 15793.41027, b6 = -8033.605502, r1 = .4599469977, r2 = .3665077232, r3 = .8087773786, r4 = .5464600340, r5 = -.8347594860, r6 = 0.6751319714e-1, r7 = -.6998786970, r8 = -.4109120215, r9 = .5842269422, t1 = -10.38583601, t2 = 10.41107518, t3 = 61.92902662],
[b1 = -9151.429703, b2 = -6008.117168, b3 = 11049.83952, b4 = -7190.606039, b5 = 13545.27447, b6 = -7223.768017, r1 = 0.7425420274e-1, r2 = .5809660735, r3 = .8105336112, r4 = -.9944330906, r5 = .1040687725, r6 = 0.1650814969e-1, r7 = -0.7476056307e-1, r8 = -.8072472434, r9 = .5854594317, t1 = -4.663325049, t2 = 4.554915592, t3 = 62.58462359],
[b1 = -1226.728760, b2 = 1147.011945, b3 = -400.7753802, b4 = -4507.321778, b5 = 8005.881456, b6 = -3241.483736, r1 = -0.9032875350e-2, r2 = .7488743127, r3 = .6626504893, r4 = -.2759411564, r5 = .6350798638, r6 = -.7214776814, r7 = .9611320853, r8 = .1893695603, r9 = -.2009086466, t1 = 40.46849908, t2 = 60.56033489, t3 = 65.08406719]]
