4 0.01692601*X1**2*Y1**2 - 0.888509280014*X1**2*Y2**2 + 0.0411717692438*X2**2*Y1**2 - 0.00437457395884*X2**2*Y2**2 + 0.331480641249*X1*X2*Y1**2 - 1.38036964668*X1*X2*Y2**2 - 0.270492270191*X1**2*Y1*Y2 + 1.44135801774*X2**2*Y1*Y2 + 0.859888946812*X1*X2*Y1*Y2 + 0.0791489659197*X1**2*Y1 - 0.00336032777032*X1**2*Y2 - 0.0620826738427*X1*Y1**2 + 0.501879647495*X1*Y2**2 + 0.647156236961*X2**2*Y1 + 0.0926311741907*X2**2*Y2 - 0.255000006226*X2*Y1**2 - 0.0896892386081*X2*Y2**2 - 0.568007271041*X1*X2*Y2 + 0.095991501961*X1*X2*Y1 + 0.165310767618*X1*Y1*Y2 - 0.563962321337*X2*Y1*Y2 - 0.0784871167595*X1*Y1 - 0.0784871167595*X2*Y2 + 0.011807283256*X1*Y2 - 0.011807283256*X2*Y1 + 0.0422876985355*X1**2 + 0.0422876985355*X2**2 + 0.0372427422943*Y1**2 + 0.0372427422943*Y2**2; 0.518178672335*X1**2*Y1**2 - 0.0414464807343*X1**2*Y2**2 + 2.63600135179*X2**2*Y1**2 - 0.799490472298*X2**2*Y2**2 + 0.29442805494*X1**2*Y1*Y2 + 1.46551534655*X2**2*Y1*Y2 - 0.631878110759*X1*X2*Y1**2 - 1.80296540237*X1*X2*Y2**2 - 2.87586667102*X1*X2*Y1*Y2 - 0.987856648177*X1**2*Y1 - 0.530579106676*X1**2*Y2 - 0.0397576281649*X1*Y1**2 + 0.317719102869*X1*Y2**2 - 1.93710490787*X2**2*Y1 + 0.00127693327315*X2**2*Y2 - 0.581380074072*X2*Y1**2 - 0.0672137066743*X2*Y2**2 + 0.531856039949*X1*X2*Y1 + 0.949248259696*X1*X2*Y2 + 0.514166367398*X1*Y1*Y2 - 0.357476731033*X2*Y1*Y2 + 0.140965913657*X1*Y1 + 0.140965913657*X2*Y2 - 0.153347218606*X1*Y2 + 0.153347218606*X2*Y1 + 0.283274882058*X1**2 + 0.283274882058*X2**2 + 0.0382903330079*Y1**2 + 0.0382903330079*Y2**2; 0.0233560008057*X1**2*Y1**2 - 0.00428427501149*X1**2*Y1*Y2 - 0.792756311827*X1**2*Y2**2 + 0.0492185850289*X2**2*Y2**2 + 0.0759264856293*X1*X2*Y1**2 + 1.14839711492*X1*X2*Y1*Y2 - 0.283066217262*X2**2*Y1**2 + 0.460041521291*X2**2*Y1*Y2 - 0.388399310674*X1*X2*Y2**2 - 0.0561169736293*X1*Y1**2 + 0.485064247792*X1*Y2**2 + 0.0689567235492*X1**2*Y1 - 0.115620658768*X1**2*Y2 - 0.13286905328*X2*Y1**2 - 0.084375901147*X2*Y2**2 + 0.639964831612*X2**2*Y1 + 0.101386684276*X2**2*Y2 + 0.217007343044*X1*X2*Y1 - 0.571008108063*X1*X2*Y2 + 0.0484931521334*X1*Y1*Y2 - 0.541181221422*X2*Y1*Y2 - 0.00363197918253*X1*Y2 + 0.00363197918253*X2*Y1 - 0.0781302968652*X1*Y1 - 0.0781302968652*X2*Y2 + 0.0471311092612*X1**2 + 0.0471311092612*X2**2 + 0.0324495575052*Y1**2 + 0.0324495575052*Y2**2; 0.393707415641*X1**2*Y1**2 + 0.59841456862*X1**2*Y2**2 + 0.0735854940135*X2**2*Y1**2 + 0.0548997238169*X2**2*Y2**2 + 0.0116156836985*X1**2*Y1*Y2 + 0.0699694273575*X2**2*Y1*Y2 - 0.305757340849*X1*X2*Y1**2 - 0.364111084508*X1*X2*Y2**2 - 0.223392923175*X1*X2*Y1*Y2 + 0.0996725944534*X1**2*Y1 + 0.0113936468426*X1**2*Y2 - 0.381205205249*X1*Y1**2 - 0.473402150235*X1*Y2**2 - 0.0213613191759*X2**2*Y1 - 0.0372595571271*X2**2*Y2 + 0.148904552394*X2*Y1**2 + 0.142408744984*X2*Y2**2 - 0.0486532039697*X1*X2*Y1 + 0.121033913629*X1*X2*Y2 - 0.00649580741066*X1*Y1*Y2 + 0.092196944986*X2*Y1*Y2 - 0.0483106652705*X1*Y1 - 0.0483106652705*X2*Y2 - 0.00316794272326*X1*Y2 + 0.00316794272326*X2*Y1 + 0.00634952598374*X1**2 + 0.00634952598374*X2**2 + 0.0922886309144*Y1**2 + 0.0922886309144*Y2**2; TITLE : a four-bar design problem, so-called 5-point problem ROOT COUNTS : total degree : 256 2-homogeneous Bezout number : 96 mixed volume : 80 REFERENCES : See Morgan, A.P. and Wampler, C.W. : `Solving a planar four-bar design problem using continuation' in Transaction of the ASME, J. of Mechanical Design, Vol. 112 pages 544-550, 1990. For the coefficients, see Table 2, a = (1,0). This is the start system, with five random precision points, that has been used to solve twenty other test systems. NOTE : At infinity, these degenerate hyperplanes are solutions of the following subsystem: 4.22876985355001E-02*X1**2+ 4.22876985355001E-02*X2**2; 2.83274882058000E-01*X1**2+ 2.83274882058000E-01*X2**2; 4.71311092612001E-02*X1**2+ 4.71311092612001E-02*X2**2; 6.34952598374001E-03*X1**2+ 6.34952598374001E-03*X2**2; There are only 36 isolated solutions, instead of 80. This problem formulation uses cartesian coordinates. By isotropic coordinates an exact root count can be obtained. THE SOLUTIONS : 36 4 =========================================================== solution 1 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : -1.30421792262534E-01 1.02275543863157E-01 Y1 : -1.34148589217653E-02 -1.76800861304744E-02 Y2 : 4.36492627551176E-02 4.26029119177140E-02 X2 : 9.30856705465082E-02 1.25957483651556E-01 == err : 3.752E-16 = rco : 2.296E-03 = res : 5.029E-19 == solution 2 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.35483057347642E-01 -6.27936033387944E-02 Y1 : 5.86610856430167E-01 -4.36257944077704E-01 Y2 : 3.82362459573002E-01 5.82597214547501E-02 X2 : 9.99310913333355E-02 2.04782353966550E-02 == err : 2.113E-15 = rco : 6.881E-03 = res : 2.861E-17 == solution 3 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.39539045851621E-01 3.37193815221795E-02 Y1 : 6.15232599153994E-01 3.18342113882502E-01 Y2 : 2.79007186501877E-01 1.34507104556481E-03 X2 : 1.02534167195379E-01 -2.14641761614356E-02 == err : 7.798E-15 = rco : 4.612E-03 = res : 1.388E-17 == solution 4 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.39539045851622E-01 -3.37193815221796E-02 Y1 : 6.15232599153994E-01 -3.18342113882501E-01 Y2 : 2.79007186501877E-01 -1.34507104556479E-03 X2 : 1.02534167195379E-01 2.14641761614357E-02 == err : 8.064E-15 = rco : 4.612E-03 = res : 2.194E-17 == solution 5 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.51313691193482E-01 -5.69342282581537E-02 Y1 : 6.78147355883156E-01 -6.22841534688354E-01 Y2 : -5.52109971919701E-02 -8.89188627716446E-02 X2 : 2.52994247634305E-02 -6.13397285290952E-02 == err : 1.148E-14 = rco : 2.017E-03 = res : 1.963E-17 == solution 6 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.41844233506457E-01 -5.78394356258555E-02 Y1 : 6.96049254825832E-01 -6.34075857297423E-01 Y2 : -2.33074644032421E-02 -9.96186647379359E-02 X2 : 3.92960662734257E-02 -5.50421790185633E-02 == err : 1.967E-14 = rco : 1.936E-03 = res : 6.960E-17 == solution 7 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.41844233506460E-01 5.78394356258546E-02 Y1 : 6.96049254825823E-01 6.34075857297422E-01 Y2 : -2.33074644032563E-02 9.96186647379346E-02 X2 : 3.92960662734207E-02 5.50421790185669E-02 == err : 1.660E-14 = rco : 1.936E-03 = res : 5.407E-17 == solution 8 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.06745321082687E-01 -1.42225069483242E-58 Y1 : 1.45199156236219E+00 -7.08995909375817E-58 Y2 : 3.68385041446078E-01 -2.18836221630536E-58 X2 : 1.62577441583587E-01 -5.27595014026165E-59 == err : 1.060E-13 = rco : 7.136E-04 = res : 1.154E-16 == solution 9 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 1.22911407793185E+00 1.15283128920095E-01 Y1 : -1.21422799853711E+00 1.94848207421399E-01 Y2 : -1.44891275906777E+00 1.11210486881801E-01 X2 : 1.58507231069686E+00 2.84084469186712E-01 == err : 7.960E-14 = rco : 1.415E-04 = res : 7.119E-15 == solution 10 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 3.03338911364224E-01 -2.20674062719568E-02 Y1 : 5.65627562106786E-01 -2.10648389346555E-01 Y2 : -4.69017514766759E-01 9.28430217877084E-02 X2 : -3.39694179390978E-01 7.35635701762810E-03 == err : 5.769E-15 = rco : 5.440E-03 = res : 2.914E-17 == solution 11 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : -1.80354055564375E-02 4.78949724993217E-02 Y1 : -6.80200292883192E-02 -2.44886147515548E-02 Y2 : 4.86848419548440E-02 -4.45456425119699E-02 X2 : 5.37892839680892E-02 5.17413048886791E-02 == err : 2.735E-16 = rco : 2.804E-03 = res : 4.371E-19 == solution 12 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 2.70729674523651E-01 -3.15111400261581E-03 Y1 : 6.01337967835043E-01 7.11392226104298E-02 Y2 : -4.35593856982499E-01 -1.70263031934523E-02 X2 : -3.72213042594878E-01 -2.10650410540472E-02 == err : 7.527E-15 = rco : 1.273E-02 = res : 2.057E-17 == solution 13 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.44692000330549E-01 2.05821216827045E-02 Y1 : 5.86421949943073E-01 3.03949700563222E-01 Y2 : 2.77119162915063E-01 4.50702056131172E-03 X2 : 1.01795437677947E-01 -2.03686721683275E-02 == err : 2.471E-14 = rco : 4.940E-03 = res : 2.861E-17 == solution 14 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.95965546075631E-01 -1.37777558251846E-57 Y1 : 1.39825982513980E+00 -6.86647505759198E-57 Y2 : 3.51809670602083E-01 -2.12267051127844E-57 X2 : 1.58592899664286E-01 -5.13359866578058E-58 == err : 8.496E-14 = rco : 8.392E-04 = res : 2.776E-17 == solution 15 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.95786845372265E-01 -9.81543760123305E-02 Y1 : 5.71041556037921E-01 5.50010099235699E-01 Y2 : 2.34674265978231E-01 -6.94090738405223E-01 X2 : 3.64967892481617E-02 5.42044962116081E-02 == err : 9.698E-16 = rco : 1.710E-02 = res : 7.356E-17 == solution 16 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 3.03338911364225E-01 2.20674062719567E-02 Y1 : 5.65627562106786E-01 2.10648389346554E-01 Y2 : -4.69017514766758E-01 -9.28430217877086E-02 X2 : -3.39694179390978E-01 -7.35635701762806E-03 == err : 6.894E-15 = rco : 5.440E-03 = res : 4.719E-17 == solution 17 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.95786845372265E-01 9.81543760123303E-02 Y1 : 5.71041556037922E-01 -5.50010099235699E-01 Y2 : 2.34674265978230E-01 6.94090738405223E-01 X2 : 3.64967892481618E-02 -5.42044962116080E-02 == err : 1.215E-15 = rco : 1.710E-02 = res : 7.755E-17 == solution 18 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.48474961174366E-01 -2.89660537402298E-02 Y1 : 4.56711896735588E-01 -5.40604010232105E-01 Y2 : 4.36650698148677E-01 1.29516750799000E-01 X2 : 9.37245901214154E-02 9.98383032374788E-03 == err : 2.705E-15 = rco : 4.647E-03 = res : 2.270E-17 == solution 19 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : -1.30421792262534E-01 -1.02275543863157E-01 Y1 : -1.34148589217653E-02 1.76800861304745E-02 Y2 : 4.36492627551178E-02 -4.26029119177140E-02 X2 : 9.30856705465084E-02 -1.25957483651556E-01 == err : 4.578E-16 = rco : 2.296E-03 = res : 1.939E-18 == solution 20 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.36525679531097E-01 6.58078823945381E-02 Y1 : 2.55591001292441E-01 -1.17965161275222E+00 Y2 : 4.00040482703310E-01 8.03172094292528E-01 X2 : 6.12863867735729E-02 -3.02901259600771E-02 == err : 2.195E-15 = rco : 6.183E-03 = res : 7.929E-17 == solution 21 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 2.70729674523652E-01 3.15111400261610E-03 Y1 : 6.01337967835043E-01 -7.11392226104275E-02 Y2 : -4.35593856982499E-01 1.70263031934518E-02 X2 : -3.72213042594879E-01 2.10650410540463E-02 == err : 6.010E-15 = rco : 1.273E-02 = res : 5.605E-17 == solution 22 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 1.22911407793185E+00 -1.15283128920080E-01 Y1 : -1.21422799853713E+00 -1.94848207421377E-01 Y2 : -1.44891275906777E+00 -1.11210486881788E-01 X2 : 1.58507231069686E+00 -2.84084469186674E-01 == err : 2.040E-13 = rco : 1.415E-04 = res : 2.232E-15 == solution 23 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 2.90599709703607E-01 1.70123717543869E-02 Y1 : 5.55565294508816E-01 2.11510701900325E-01 Y2 : -4.68186951883872E-01 -9.77440998333686E-02 X2 : -3.35488612100868E-01 -6.01260393173532E-03 == err : 4.574E-15 = rco : 5.194E-03 = res : 2.412E-17 == solution 24 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.44692000330550E-01 -2.05821216827044E-02 Y1 : 5.86421949943074E-01 -3.03949700563217E-01 Y2 : 2.77119162915063E-01 -4.50702056131171E-03 X2 : 1.01795437677947E-01 2.03686721683275E-02 == err : 4.691E-15 = rco : 4.940E-03 = res : 3.103E-17 == solution 25 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.02136933783012E-01 -3.88468477571993E-02 Y1 : 1.89744852767543E+00 9.52624110251982E-02 Y2 : -3.58421896794606E-01 1.07330408787373E-01 X2 : 1.59778295802974E-01 -5.65382606893851E-02 == err : 3.785E-15 = rco : 3.015E-02 = res : 2.231E-16 == solution 26 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 2.98081022796265E-01 -5.59177793201383E-03 Y1 : 6.08989767717123E-01 -9.03498053632811E-02 Y2 : -4.38183255792341E-01 2.12961599038053E-02 X2 : -3.73213180683383E-01 2.43142377285910E-02 == err : 1.220E-14 = rco : 1.136E-02 = res : 4.995E-17 == solution 27 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.51313691193481E-01 5.69342282581517E-02 Y1 : 6.78147355883155E-01 6.22841534688358E-01 Y2 : -5.52109971919685E-02 8.89188627716519E-02 X2 : 2.52994247634334E-02 6.13397285290973E-02 == err : 1.760E-14 = rco : 2.017E-03 = res : 2.809E-17 == solution 28 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 2.98081022796264E-01 5.59177793201374E-03 Y1 : 6.08989767717121E-01 9.03498053632746E-02 Y2 : -4.38183255792340E-01 -2.12961599038041E-02 X2 : -3.73213180683385E-01 -2.43142377285892E-02 == err : 4.733E-15 = rco : 1.136E-02 = res : 1.703E-17 == solution 29 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.13486066538937E-01 -5.05101069868295E-03 Y1 : 1.85992750826152E+00 -4.33106907379951E-02 Y2 : -3.44556622517745E-01 -5.27625248166793E-02 X2 : 1.64005238422230E-01 6.00581327972965E-02 == err : 4.761E-15 = rco : 5.764E-02 = res : 2.227E-16 == solution 30 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.13486066538937E-01 5.05101069868323E-03 Y1 : 1.85992750826152E+00 4.33106907379940E-02 Y2 : -3.44556622517745E-01 5.27625248166790E-02 X2 : 1.64005238422230E-01 -6.00581327972965E-02 == err : 3.706E-15 = rco : 5.764E-02 = res : 1.111E-16 == solution 31 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.02136933783012E-01 3.88468477571977E-02 Y1 : 1.89744852767542E+00 -9.52624110251948E-02 Y2 : -3.58421896794605E-01 -1.07330408787371E-01 X2 : 1.59778295802974E-01 5.65382606893851E-02 == err : 5.554E-15 = rco : 3.015E-02 = res : 1.122E-16 == solution 32 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 5.36525679531096E-01 -6.58078823945380E-02 Y1 : 2.55591001292440E-01 1.17965161275222E+00 Y2 : 4.00040482703312E-01 -8.03172094292525E-01 X2 : 6.12863867735729E-02 3.02901259600771E-02 == err : 3.832E-15 = rco : 6.183E-03 = res : 1.074E-16 == solution 33 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.35483057347643E-01 6.27936033387947E-02 Y1 : 5.86610856430169E-01 4.36257944077703E-01 Y2 : 3.82362459573001E-01 -5.82597214547501E-02 X2 : 9.99310913333354E-02 -2.04782353966552E-02 == err : 3.517E-15 = rco : 6.881E-03 = res : 1.289E-17 == solution 34 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 4.48474961174366E-01 2.89660537402296E-02 Y1 : 4.56711896735587E-01 5.40604010232105E-01 Y2 : 4.36650698148676E-01 -1.29516750799000E-01 X2 : 9.37245901214154E-02 -9.98383032374780E-03 == err : 4.057E-15 = rco : 4.647E-03 = res : 3.133E-17 == solution 35 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : 2.90599709703608E-01 -1.70123717543856E-02 Y1 : 5.55565294508815E-01 -2.11510701900326E-01 Y2 : -4.68186951883870E-01 9.77440998333699E-02 X2 : -3.35488612100867E-01 6.01260393173520E-03 == err : 5.620E-15 = rco : 5.194E-03 = res : 2.453E-17 == solution 36 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : X1 : -1.80354055564375E-02 -4.78949724993217E-02 Y1 : -6.80200292883192E-02 2.44886147515548E-02 Y2 : 4.86848419548440E-02 4.45456425119699E-02 X2 : 5.37892839680892E-02 -5.17413048886791E-02 == err : 2.735E-16 = rco : 2.804E-03 = res : 4.371E-19 ==