2019-03-03 16:01:58 -0600 received badge ● Notable Question (source) 2019-02-11 19:27:55 -0600 received badge ● Notable Question (source) 2018-03-29 07:58:39 -0600 received badge ● Popular Question (source) 2018-03-07 07:44:54 -0600 received badge ● Popular Question (source) 2018-01-01 07:15:44 -0600 commented answer Getting the error message: The dimension of the ideal is 1, but it should be 0 I will look at it thanks. It was just guess. The reason I said that is that if I write: I = Ideal(eq1) I.dimension() I get 4. and if I define the ideal with 2 equations the dimension is 3 and so on. 2017-12-31 20:12:13 -0600 answered a question Getting the error message: The dimension of the ideal is 1, but it should be 0 I actually understood my mistake. I accidently wrote twice the same equation (eq2=-eq3). So I guess that the meaning of the dimension of ideal (at least in this case) is the number of equations that are linear combination of the other equations (therefore should be zero and not positive number). I'm writing this here in case anyone else gets the same error in the future. 2017-12-31 16:24:09 -0600 asked a question Getting the error message: The dimension of the ideal is 1, but it should be 0 I'm trying to solve a system of 5 polynomial equations. Here's the code I tried to run: P.=PolynomialRing(QQ,order='degrevlex') eq1=P(-25997.02495*qc+73589.75314*qa+19275.89428*qb^3+42024.09724*qc^3-35275.79436*qd^3+31409.96375*qd-22475.53767*qb+11165.49567*qc*qa*qd+38392.81504*qd*qb*qc-5354.736466*qc*qa*qb-40769.13796*qd*qb*qa-50708.36034*qc^2*qa-67780.84581*qd*qb^2+39326.95066*qd^2*qb+5359.28038*qb^2*qa-35437.12402*qd*qc^2+48529.90789*qd^2*qc-6801.32966*qb^2*qc+13747.85604*qc^2*qb+9197.21841*qd^2*qa+2*b*qa) eq2=-P(1938.516702*qc-9153.752714*qa-15279.24300*qb^3+8131.520743*qc^3+35208.27094*qd^3-25334.07110*qd+71321.57867*qb+38392.81500*qc*qa*qd-48554.44832*qd*qb*qc-64028.43710*qc*qa*qb-58696.44966*qd*qb*qa+2*b*qb+426.071103*qc^2*qa+26980.57949*qd*qb^2-29944.11326*qd^2*qb+4540.542985*qb^2*qa-1130.66934*qd*qc^2-8809.128406*qd^2*qc-9514.136166*qb^2*qc-21010.04302*qc^2*qb+26005.16567*qd^2*qa) eq3=P(1938.516702*qc-9153.752714*qa-15279.24300*qb^3+8131.520743*qc^3+35208.27094*qd^3-25334.07110*qd+71321.57867*qb+38392.81500*qc*qa*qd-48554.44832*qd*qb*qc-64028.43710*qc*qa*qb-58696.44966*qd*qb*qa+2*b*qb+426.071103*qc^2*qa+26980.57949*qd*qb^2-29944.11326*qd^2*qb+4540.542985*qb^2*qa-1130.66934*qd*qc^2-8809.128406*qd^2*qc-9514.136166*qb^2*qc-21010.04302*qc^2*qb+26005.16567*qd^2*qa) eq4=P(2123.141846*qc+5788.216407*qa+22583.23914*qb^3-8500.597767*qc^3-9489.15234*qd^3+75013.55778*qd-25334.07110*qb+46634.03800*qc*qa*qd-22972.99327*qd*qb*qc+38392.81504*qc*qa*qb+38688.54642*qd*qb*qa+2*b*qd-9815.37666*qc^2*qa-33782.05127*qd*qb^2+64855.67483*qd^2*qb-42159.09845*qb^2*qa-35072.07256*qd*qc^2-8694.617828*qd^2*qc-29859.97200*qb^2*qc-1130.66933*qc^2*qb-3340.393775*qd^2*qa) eq5= P(qa^2+qb^2+qc^2+qd^2-1) I = Ideal(eq1, eq2, eq3, eq4, eq5) I.groebner_basis('libsingular:std') I.variety(RR)  At the last line of code: I.variety(RR)  I get this error: The dimension of the ideal is 1, but it should be 0  Anyone knows why? or alternatively, how to get the solutions of the equations? Thanks 2017-12-28 20:25:53 -0600 asked a question Solving a system of 5 polynomial equations I have a system of 5 polynomial equations. I solved the equations in Maple but I'm trying to solve them also in sagemath since it's open source. I've been trying for a long time (weeks) and still couldn't solve the equations in sagemath (Maple finds all the solutions in 10 seconds). I don't need the complex solutions (if exist). only the real solutions. I have posted before a similar question but with 18 equations and it took Maple 40 minutes to solve them. But since then I was able to reduce the number of equations from 18 to 5. This time Maple solved the equations much faster (10 seconds instead of 40 minutes). But unfortunately, still can't solve the equations with sagemath. The goal is to find all the possible (real) solutions of the equations. Maple did it and found 20 solutions (the code is attached bellow). But I wasn't able to solve the equations in sagemath at all (I attached the code for sagemath bellow as well). I tried to use groebner basis in sagemath but there are so many different implementations with different order of the polynomials ('lex', 'degrevlex',etc). I tried many of them and none of them could solve the equations. I will appreciate if anyone can suggest other ways to solve the equations. Maybe not groebner? any method that works and returns all the 20 solutions. This is the code in Maple that defines the 5 equations and finds all the solutions: restart; eq1:=-18889.48706*qd+467.29186*qb+2982.844413*qd*qc*qb+30136.14351*qc-1115.07186*qc^2*qd-45629.75749*qa*qd*qb-23697.88597*qd^2*qc+4316.66628*qa*qc*qb-15135.14056*qb^2*qc-21851.63993*qa*qd*qc+7902.042467*qb^2*qd-5483.40637*qd^2*qb-11776.45729*qc^2*qb+40621.75090*qa+11348.58548*qd^3-18144.72833*qb^3-19544.23125*qc^3+2*b*qa+32521.65211*qa*qc^2+25623.87525*qa*qd^2+26225.65173*qa*qb^2; eq2:=-29579.52853*qd+73144.02842*qb+8748.09605*qd*qc*qb+2240.035087*qc+26343.71460*qc^2*qd+2344.316752*qa*qd*qb-25331.91138*qd^2*qc+1444.808993*qa*qc*qb+6290.87549*qb^2*qc+2982.844337*qa*qd*qc+49189.98407*qb^2*qd-29432.25353*qd^2*qb-17559.89085*qc^2*qb-7338.23254*qa+2*b*qb+21944.51817*qd^3-11930.76271*qb^3+2394.88100*qc^3-3970.93290*qa*qc^2+2322.11803*qa*qd^2-23212.08740*qa*qb^2; eq3:=-3375.795773*qd+2240.03510*qb+7057.67167*qd*qc*qb+56197.14569*qc+52577.03189*qc^2*qd+2982.84431*qa*qd*qb-17977.24342*qd^2*qc-136.341409*qa*qc*qb-23855.89112*qb^2*qc-15689.91200*qa*qd*qc+15299.86799*qb^2*qd-25331.91138*qd^2*qb+11501.30926*qc^2*qb+19564.44679*qa-963.74805*qd^3+658.069730*qb^3-24654.25363*qc^3-16345.90692*qa*qc^2-13126.18932*qa*qd^2-4563.44385*qa*qb^2+2*b*qc; eq4:=37246.67872*qd-29579.52855*qb-46347.15644*qd*qc*qb-3375.795773*qc-11079.46666*qc^2*qd+12449.76046*qa*qd*qb-24742.88407*qd^2*qc+2982.844222*qa*qc*qb+15299.86800*qb^2*qc-15680.68187*qa*qd*qc-28830.47715*qb^2*qd+20203.79697*qd^2*qb+26343.71462*qc^2*qb-14402.89763*qa-14953.58829*qd^3+31606.58056*qb^3+24809.55728*qc^3-5601.66129*qa*qc^2+16099.39874*qa*qd^2+3415.45305*qa*qb^2+2*b*qd; eq5:=qa^2+qb^2+qc^2+qd^2-1; vars:=[b, qa, qb, qc, qd]; polysys:={eq1,eq2,eq3,eq4,eq5}; sols:=CodeTools:-Usage(RootFinding:-Isolate(polysys, vars, output = numeric, method = RS));  These are all the solutions that Maple found (20 solutions): sols := [[b = -6104.435348, qa = -.2144080500, qb = -.2880353207, qc = -.1836576407, qd = -.9150599506], [b = -12366.14419, qa = -.5402828747, qb = -.3634451559, qc = .1841090309, qd = -.7362784111], [b = -24212.37926, qa = .2546299302, qb = .6836963191, qc = -.4499174642, qd = -.5150701091], [b = -23961.92853, qa = .2585879982, qb = .7021280140, qc = -.4278033463, qd = -.5070826324], [b = -6007.960508, qa = -.3010769760, qb = -.1196072448, qc = .8107192013, qd = -.4876280735], [b = -22102.25884, qa = .7018528580, qb = -.1525453855, qc = -.5362892523, qd = -.4433128793], [b = -26791.96626, qa = .3384082623, qb = -.1163579699, qc = -.8287561155, qd = -.4302371112], [b = -26003.78615, qa = 0.7746836766e-1, qb = -.1265934010, qc = -.9030243373, qd = -.4031374569], [b = -27290.86178, qa = .3563297191, qb = .8341751821, qc = .4195778650, qd = -0.3369439196e-1], [b = -39895.97736, qa = -.5758752754, qb = .7983140794, qc = -.1758240948, qd = -0.1217314478e-1], [b = -39895.97736, qa = .5758752754, qb = -.7983140794, qc = .1758240948, qd = 0.1217314478e-1], [b = -27290.86178, qa = -.3563297191, qb = -.8341751821, qc = -.4195778650, qd = 0.3369439196e-1], [b = -26003.78615, qa = -0.7746836766e-1, qb = .1265934010, qc = .9030243373, qd = .4031374569], [b = -26791.96626, qa = -.3384082623, qb = .1163579699, qc = .8287561155, qd = .4302371112], [b = -22102.25884, qa = -.7018528580, qb = .1525453855, qc = .5362892523, qd = .4433128793], [b = -6007.960508, qa = .3010769760, qb = .1196072448, qc = -.8107192013, qd = .4876280735], [b = -23961.92853, qa = -.2585879982, qb = -.7021280140, qc = .4278033463, qd = .5070826324], [b = -24212.37926, qa = -.2546299302, qb = -.6836963191, qc = .4499174642, qd = .5150701091], [b = -12366.14419, qa = .5402828747, qb = .3634451559, qc = -.1841090309, qd = .7362784111], [b = -6104.435348, qa = .2144080500, qb = .2880353207, qc = .1836576407, qd = .9150599506]]  And this is the code in sagemath that I couldn't solve. Same equations as before, only converted them to rational coefficients using eq_i=P(eq_i). It is stuck in the last line of code (I.variety(RR)). P.=PolynomialRing(QQ,order='degrevlex') eq1=-18889.48706*qd+467.29186*qb+2982.844413*qd*qc*qb+30136.14351*qc-1115.07186*qc^2*qd-45629.75749*qa*qd*qb-23697.88597*qd^2*qc+4316.66628*qa*qc*qb-15135.14056*qb^2*qc-21851.63993*qa*qd*qc+7902.042467*qb^2*qd-5483.40637*qd^2*qb-11776.45729*qc^2*qb+40621.75090*qa+11348.58548*qd^3-18144.72833*qb^3-19544.23125*qc^3+2*b*qa+32521.65211*qa*qc^2+25623.87525*qa*qd^2+26225.65173*qa*qb^2 eq2=-29579.52853*qd+73144.02842*qb+8748.09605*qd*qc*qb+2240.035087*qc+26343.71460*qc^2*qd+2344.316752*qa*qd*qb-25331.91138*qd^2*qc+1444.808993*qa*qc*qb+6290.87549*qb^2*qc+2982.844337*qa*qd*qc+49189.98407*qb^2*qd-29432.25353*qd^2*qb-17559.89085*qc^2*qb-7338.23254*qa+2*b*qb+21944.51817*qd^3-11930.76271*qb^3+2394.88100*qc^3-3970.93290*qa*qc^2+2322.11803*qa*qd^2-23212.08740*qa*qb^2 eq3=-3375.795773*qd+2240.03510*qb+7057.67167*qd*qc*qb+56197.14569*qc+52577.03189*qc^2*qd+2982.84431*qa*qd*qb-17977.24342*qd^2*qc-136.341409*qa*qc*qb-23855.89112*qb^2*qc-15689.91200*qa*qd*qc+15299.86799*qb^2*qd-25331.91138*qd^2*qb+11501.30926*qc^2*qb+19564.44679*qa-963.74805*qd^3+658.069730*qb^3-24654.25363*qc^3-16345.90692*qa*qc^2-13126.18932*qa*qd^2-4563.44385*qa*qb^2+2*b*qc eq4=37246.67872*qd-29579.52855*qb-46347.15644*qd*qc*qb-3375.795773*qc-11079.46666*qc^2*qd+12449.76046*qa*qd*qb-24742.88407*qd^2*qc+2982.844222*qa*qc*qb+15299.86800*qb^2*qc-15680.68187*qa*qd*qc-28830.47715*qb^2*qd+20203.79697*qd^2*qb+26343.71462*qc^2*qb-14402.89763*qa-14953.58829*qd^3+31606.58056*qb^3+24809.55728*qc^3-5601.66129*qa*qc^2+16099.39874*qa*qd^2+3415.45305*qa*qb^2+2*b*qd eq5=qa^2+qb^2+qc^2+qd^2-1 eq1=P(eq1) eq2=P(eq2) eq3=P(eq3) eq4=P(eq4) eq5=P(eq5) I=Ideal(eq1, eq2, eq3, eq4, eq5) I.groebner_basis('libsingular:std') I.variety(RR)  Anyone knows how to make it work? Thanks 2017-12-11 08:29:26 -0600 received badge ● Commentator 2017-12-11 08:29:26 -0600 commented answer Solving a system of 18 polynomial equations in sagemath Do you mean that I should make the coefficients integer numbers? doesn't it mean that it will find only the integer solutions (which most likely means empty set)? if some groebner algorithms work only with ZZ shouldn't I have gotten an error message? If it can find real solutions with ZZ ring, can you show me how to convert the expression to expression with integer coefficients? I'm very new with sagemath so I don't know how to collect all the denominators of the coefficients of an expression so I can compute their LCM. Thanks. 2017-12-11 04:51:08 -0600 commented answer Solving a system of 18 polynomial equations in sagemath I already tried a few different algorithms of groebner. I tried ‘singular:groebner’, ‘singular:std’, ‘singular:stdhilb’, ‘singular:stdfglm’ and some more from this link: http://doc.sagemath.org/html/en/refer.... None of them worked. They all never finished. for the specific algorithm you wrote 'giac:gbasis' I couldn't run. I got this error: "One of the optional packages giac or giacpy_sage is missing". I don't know how to install the missing package. I also tried 'lex' instead of 'degrevlex'. also didn't work. Is there any other idea? perhaps not using QQ? it doesn't matter for me if it's rational numbers or real numbers. whatever works. 2017-12-10 16:13:06 -0600 commented answer Solving a system of 18 polynomial equations in sagemath I updated the original question with the updated code (just changed sum_sqr_distances to be rational). Unfortunately it seems that I.groebner_basis() is not finishing. Is there any other way that I can try to calculate the groebner basis of the 18 equations? about the link that you gave me for optimizations, it seems that I need to give some guesses to the function so it will look for the local minimum around these values (or interval in another function). This is definitely not what I want. I need a global minimum for the problem that is guarantees to be the global minimum (with no guesses). 2017-12-10 13:20:13 -0600 received badge ● Supporter (source) 2017-12-10 13:19:54 -0600 commented answer Solving a system of 18 polynomial equations in sagemath The complete input of my problem is just n random points and n random lines. each line consists of a random point and a random unit vector. the random points (and also the points on the lines) are random real numbers (no square roots). but since the vectors are unit vectors, I divide a random vector by its size so obviously I use square root to define a random unit vector ((vx,vy,vz)/sqrt(vx^2+vy^2+vz^2)). Why should I forget about polynomial equations? it worked in Maple. can you tell me which function you were talking about in the link you sent? because I tried many optimization functions in Maple and none of them guranteed to return the global minimum. finding all the solutions of the polynomial equations guarantees that (and that was the only way I could solve the problem in Maple). 2017-12-10 13:14:10 -0600 commented answer Solving a system of 18 polynomial equations in sagemath Thanks. I didn't notice that. so my code now is exactly the same as in the original question (still defined P with QQ). just added after the definition of sum_sqr_distances the line: "sum_sqr_distances=P(sum_sqr_distances)". Now the command I.groebner_basis() is running for almost 2 hours (didn't return [1.0] this time). but I'm not sure it will ever finish.. 2017-12-10 10:24:43 -0600 commented answer Solving a system of 18 polynomial equations in sagemath Is there a way to replace the expressions to be with rational coefficients? 2017-12-10 10:19:20 -0600 commented answer Solving a system of 18 polynomial equations in sagemath I tried to replace QQ with both RDF and RR. with both ways I still got a single groebner basis=1.0. Define what in QQbar? I'm not sure I understand 2017-12-10 06:23:29 -0600 commented question Solving a system of 18 polynomial equations in sagemath I updated my question so it should be clearer. The variables are now shorter. and I also wrote in the question the mathematical problem that I'm trying to solve. to your question, it is possible to illustrate the same problem with less variables. You can illustrate it with 12 polynomial equations (instead of 18) but the polynomials will be of higher degree (degree 6 instead of 2). which makes it impossible even for Maple to solve it. You can also illustrate it with 6 variables (3 angles of rotation and 3 variables for translation) but then the equations will be very long and ugly trigonometric equations and most likely unsolvable. The only way I could get a solution from Maple which if for sure correct is with 18 polynoimal equations of second degree. 2017-12-08 09:57:04 -0600 commented question Solving a system of 18 polynomial equations in sagemath I'm not sure what you meant by simplifying the example. All I did in the code examples (in both maple and sage) was to define the 18 polynomial equations and then calling the commands that solve them (solved successfully in Maple but not in sage). Let me know what you meant and I'll change my post accordingly. 2017-12-08 09:22:16 -0600 received badge ● Student (source) 2017-12-07 19:25:12 -0600 commented question Solving a system of 18 polynomial equations in sagemath Sorry I didn't notice that it didn't copy the sage code properly (I could see the correct code in the edit mode but on the preview mode the P ring was not defined properly). Anyway, I fixed it and used the ctrl+k as you said. I don't see any problem with multiplication signs (maybe that was again only in the preview mode before I pressed the ctrl+k). you can just copy the code as is, save it in as a sage file and then load the file in sagemath. it works only until the command I.groebner_basis(). 2017-12-07 18:13:12 -0600 received badge ● Editor (source) 2017-12-07 18:02:47 -0600 asked a question 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 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 22 solutions to the equations (I wrote below all the 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 but the command I.groebner_basis() never finishes (in the example below). Is there maybe another way of getting the groebner basis of the 18 equations? or any other way of solving the problem (I wrote below) so that the solution is for sure correct? 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 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 18 variables (equal to zero). it is gurantee that one of the solutions to the 18 equations is the (global) minimum for R and T. but for that I have to find ALL the solutions to the equations (22 solutions in this example) just like Maple did. Thanks Here is Maple code: (and below sage's code) restart; 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:={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));  And this is sage code (that couldn't find the groebner basis): P.=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+37.0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sum_sqr_distances=P(sum_sqr_distances) 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(diff_t1,diff_t2,diff_t3,diff_r1,diff_r2,diff_r3,diff_r4,diff_r5,diff_r6,diff_r7,diff_r8,diff_r9,diff_b1,diff_b2,diff_b3,diff_b4,diff_b5,diff_b6) I.groebner_basis() I.variety(RR)  And these are the 22 solutions the Maple found (all the real solutions that solve the equations): The goal is to get the same solutions from sage. [[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]]