var x1;
var x2;
var x3;
var x4;
var x5;
var x6;
var x7;
var x8;

subject to

cons1:  x1^2 + x2^2 - 1 = 0;
cons2:  x3^2 + x4^2 - 1 = 0;
cons3:  x5^2 + x6^2 - 1 = 0;
cons4:  x7^2 + x8^2 - 1 = 0;
cons5:  0.004731*x1*x3 - 0.3578*x2*x3 - 0.1238*x1 - 0.001637*x2 - 0.9338*x4 + x7 
- 0.3571 = 0;
cons6:  0.2238*x1*x3 + 0.7623*x2*x3 + 0.2638*x1 - 0.07745*x2 -0.6734*x4 -0.6022 = 0;
cons7:  x6*x8 + 0.3578*x1 + 0.004731*x2 = 0;
cons8:  -0.7623*x1 + 0.2238*x2 + 0.3461 = 0;

solve;
display x1, x2, x3, x4, x5, x6, x7, x8;

# TITLE : hand position and orientation of PUMA robot

# ROOT COUNTS :

# total degree : 128
# 3-homogeneous Bezout number : 16
#   with partition {{x1 x2 }{x3 x4 x7 x8 }{x5 x6 }}
# mixed volume : 16

# REFERENCES :

# A. Morgan and V. Shapiro.
#  `Box-Bisection for solving second-degree systems and the problem
#   of clustering', ACM Transactions on Mathematical Software,
#  Vol. 13, No. 2, pp. 152-167, 1987.

# NOTE :

# The system can be easily reduced to a lower dimensional one,
# by eliminating x1 or x2 by the last equation.
# It is not a hard problem to solve, but a very good test example
# for algorithms that verify whether the mixed volume is zero or not.
# Every fine mixed subdivision will have only one mixed cell.
# More precisely, there are 6336 different configurations, which all
# yield a mixed cell with mixed volume 16.

# THE SOLUTIONS :

# 16 8
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   8.56002160304148E-55
#  x2 : -9.86388476850967E-01   2.61012178719941E-54
#  x3 :  2.39616017204621E-01   5.94496235550677E-54
#  x4 : -9.70867737799024E-01   1.46725258920162E-54
#  x5 :  9.97635398227387E-01   2.66040587238087E-52
#  x6 :  6.87285399501796E-02  -3.86173644000454E-51
#  x7 : -6.15508407155080E-01   0.00000000000000E+00
#  x8 : -7.88130319630844E-01  -4.42836640644059E-50
# == err :  5.847E-16 = rco :  1.742E-01 = res :  1.631E-16 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   8.56002160304148E-55
#  x2 : -9.86388476850967E-01   2.61012178719941E-54
#  x3 :  2.39616017204621E-01   5.94496235550677E-54
#  x4 : -9.70867737799024E-01   1.46725258920162E-54
#  x5 : -9.97635398227387E-01  -8.57311908487996E-50
#  x6 : -6.87285399501796E-02   2.56585412168851E-49
#  x7 : -6.15508407155080E-01   0.00000000000000E+00
#  x8 :  7.88130319630844E-01  -2.10522022959584E-49
# == err :  5.847E-16 = rco :  2.040E-01 = res :  1.631E-16 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   8.56002160304148E-55
#  x2 : -9.86388476850967E-01   2.61012178719941E-54
#  x3 :  2.39616017204621E-01   5.94496235550677E-54
#  x4 : -9.70867737799024E-01   1.46725258920162E-54
#  x5 : -9.97635398227387E-01  -2.66040587238087E-52
#  x6 :  6.87285399501796E-02  -3.86173644000454E-51
#  x7 : -6.15508407155080E-01   0.00000000000000E+00
#  x8 : -7.88130319630844E-01  -4.42836640644059E-50
# == err :  5.847E-16 = rco :  2.040E-01 = res :  1.631E-16 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   8.56002160304148E-55
#  x2 : -9.86388476850967E-01   2.61012178719941E-54
#  x3 :  2.39616017204621E-01   5.94496235550677E-54
#  x4 : -9.70867737799024E-01   1.46725258920162E-54
#  x5 :  9.97635398227387E-01   8.57311908487996E-50
#  x6 : -6.87285399501796E-02   2.56585412168851E-49
#  x7 : -6.15508407155080E-01   0.00000000000000E+00
#  x8 :  7.88130319630844E-01  -2.10522022959584E-49
# == err :  5.847E-16 = rco :  1.742E-01 = res :  1.631E-16 ==
# solution 5 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   0.00000000000000E+00
#  x2 : -9.86388476850967E-01   2.51345585423244E-88
#  x3 : -9.54728434485888E-01   4.11804607157442E-83
#  x4 :  2.97478766274377E-01   0.00000000000000E+00
#  x5 :  9.11154796098708E-01   7.46761833343337E-60
#  x6 : -4.12064239586890E-01   3.48488855560224E-59
#  x7 :  9.91322415106624E-01   3.89318277576152E-64
#  x8 :  1.31452916693283E-01   3.69491532123005E-61
# == err :  5.116E-15 = rco :  9.651E-03 = res :  1.110E-16 ==
# solution 6 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   0.00000000000000E+00
#  x2 : -9.86388476850967E-01   2.51345585423244E-88
#  x3 : -9.54728434485888E-01   4.11804607157442E-83
#  x4 :  2.97478766274377E-01   0.00000000000000E+00
#  x5 : -9.11154796098708E-01  -7.46761833343337E-60
#  x6 :  4.12064239586890E-01  -3.48488855560224E-59
#  x7 :  9.91322415106624E-01   3.89318277576152E-64
#  x8 : -1.31452916693283E-01  -3.69491532123005E-61
# == err :  5.116E-15 = rco :  9.729E-03 = res :  1.110E-16 ==
# solution 7 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   0.00000000000000E+00
#  x2 : -9.86388476850967E-01   2.51345585423244E-88
#  x3 : -9.54728434485888E-01   4.11804607157442E-83
#  x4 :  2.97478766274377E-01   0.00000000000000E+00
#  x5 : -9.11154796098708E-01  -7.46761833343337E-60
#  x6 : -4.12064239586890E-01   3.48488855560224E-59
#  x7 :  9.91322415106624E-01   3.89318277576152E-64
#  x8 :  1.31452916693283E-01   3.69491532123005E-61
# == err :  5.116E-15 = rco :  9.729E-03 = res :  1.110E-16 ==
# solution 8 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  1.64431665854327E-01   0.00000000000000E+00
#  x2 : -9.86388476850967E-01   2.51345585423244E-88
#  x3 : -9.54728434485888E-01   4.11804607157442E-83
#  x4 :  2.97478766274377E-01   0.00000000000000E+00
#  x5 :  9.11154796098708E-01   7.46761833343337E-60
#  x6 :  4.12064239586890E-01  -3.48488855560224E-59
#  x7 :  9.91322415106624E-01   3.89318277576152E-64
#  x8 : -1.31452916693283E-01  -3.69491532123005E-61
# == err :  5.116E-15 = rco :  9.651E-03 = res :  1.110E-16 ==
# solution 9 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.08809742975953E-53
#  x2 :  7.40955378840649E-01   2.50571691571143E-52
#  x3 : -2.39611659195710E-01   2.77934733962389E-52
#  x4 : -9.70868813371549E-01   0.00000000000000E+00
#  x5 : -9.57917101877785E-01  -1.04404871487976E-53
#  x6 : -2.87044989383310E-01   4.17619485951906E-53
#  x7 : -5.27909026365623E-01   8.35238971903811E-53
#  x8 :  8.49300924220444E-01   8.35238971903811E-53
# == err :  5.564E-16 = rco :  1.733E-01 = res :  1.110E-16 ==
# solution 10 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.08809742975953E-53
#  x2 :  7.40955378840649E-01   2.50571691571143E-52
#  x3 : -2.39611659195710E-01   2.50571691571143E-52
#  x4 : -9.70868813371549E-01  -8.35238971903811E-53
#  x5 :  9.57917101877785E-01   6.26429228927858E-53
#  x6 :  2.87044989383310E-01  -2.08809742975953E-52
#  x7 : -5.27909026365623E-01   0.00000000000000E+00
#  x8 : -8.49300924220444E-01  -8.35238971903811E-53
# == err :  5.564E-16 = rco :  1.667E-01 = res :  1.110E-16 ==
# solution 11 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.08809742975953E-53
#  x2 :  7.40955378840649E-01   2.50571691571143E-52
#  x3 : -2.39611659195710E-01   2.77934733962389E-52
#  x4 : -9.70868813371549E-01   0.00000000000000E+00
#  x5 :  9.57917101877785E-01   1.04404871487976E-53
#  x6 : -2.87044989383310E-01   4.17619485951906E-53
#  x7 : -5.27909026365623E-01   8.35238971903811E-53
#  x8 :  8.49300924220444E-01   8.35238971903811E-53
# == err :  5.564E-16 = rco :  1.667E-01 = res :  1.110E-16 ==
# solution 12 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.08809742975953E-53
#  x2 :  7.40955378840649E-01   2.50571691571143E-52
#  x3 : -2.39611659195710E-01   2.50571691571143E-52
#  x4 : -9.70868813371549E-01  -8.35238971903811E-53
#  x5 : -9.57917101877785E-01  -6.26429228927858E-53
#  x6 :  2.87044989383310E-01  -2.08809742975953E-52
#  x7 : -5.27909026365623E-01   0.00000000000000E+00
#  x8 : -8.49300924220444E-01  -8.35238971903811E-53
# == err :  5.564E-16 = rco :  1.733E-01 = res :  1.110E-16 ==
# solution 13 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.10843958864610E-81
#  x2 :  7.40955378840649E-01  -6.32531876593832E-81
#  x3 :  9.54729769791417E-01  -1.76673203694632E-80
#  x4 :  2.97474480710579E-01   3.37350334183377E-80
#  x5 : -1.28778236227690E-01   2.59425224673324E-49
#  x6 :  9.91673416944351E-01  -5.54759771060745E-71
#  x7 :  9.69311807770115E-01   2.26391976970668E-72
#  x8 : -2.45834536461890E-01  -1.37524217994990E-71
# == err :  5.075E-14 = rco :  3.169E-03 = res :  1.110E-16 ==
# solution 14 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.10843958864610E-81
#  x2 :  7.40955378840649E-01  -6.32531876593832E-81
#  x3 :  9.54729769791417E-01  -1.76673203694632E-80
#  x4 :  2.97474480710579E-01   3.37350334183377E-80
#  x5 :  1.28778236227690E-01  -2.59425224673324E-49
#  x6 : -9.91673416944351E-01   5.54759771060745E-71
#  x7 :  9.69311807770115E-01   2.26391976970668E-72
#  x8 :  2.45834536461890E-01   1.37524217994990E-71
# == err :  5.075E-14 = rco :  3.187E-03 = res :  1.110E-16 ==
# solution 15 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.10843958864610E-81
#  x2 :  7.40955378840649E-01  -6.32531876593832E-81
#  x3 :  9.54729769791417E-01  -1.76673203694632E-80
#  x4 :  2.97474480710579E-01   3.37350334183377E-80
#  x5 :  1.28778236227690E-01  -2.59425224673324E-49
#  x6 :  9.91673416944351E-01  -5.54759771060745E-71
#  x7 :  9.69311807770115E-01   2.26391976970668E-72
#  x8 : -2.45834536461890E-01  -1.37524217994990E-71
# == err :  5.075E-14 = rco :  3.187E-03 = res :  1.110E-16 ==
# solution 16 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x1 :  6.71554261818887E-01   2.10843958864610E-81
#  x2 :  7.40955378840649E-01  -6.32531876593832E-81
#  x3 :  9.54729769791417E-01  -1.76673203694632E-80
#  x4 :  2.97474480710579E-01   3.37350334183377E-80
#  x5 : -1.28778236227690E-01   2.59425224673324E-49
#  x6 : -9.91673416944351E-01   5.54759771060745E-71
#  x7 :  9.69311807770115E-01   2.26391976970668E-72
#  x8 :  2.45834536461890E-01   1.37524217994990E-71
# == err :  5.075E-14 = rco :  3.169E-03 = res :  1.110E-16 ==