6
 x0**2 + x1**2 + x2**2 - 1;
 x0*x3 + x1*x4 + x2*x5;
 x3**2 + x4**2 + x5**2 - 0.25;
 x3**2 + x4**2 - 2*x2*x4 + x2**2 + x5**2 + 2*x1*x5 + x1**2 - 0.25;

 x3**2 + 1.73205080756888*x2*x3 + 0.75*x2**2 + x4**2 - x2*x4 + 0.25*x2**2
 + x5**2 - 1.73205080756888*x0*x5 + x1*x5
 + 0.75*x0**2 - 0.86602540378444*x0*x1 + 0.25*x1**2 - 0.25;

 x3**2 - 1.63299316185545*x1*x3 + 0.57735026918963*x2*x3
 + 0.66666666666667*x1**2 - 0.47140452079103*x1*x2 + 0.08333333333333*x2**2

 + x4**2 + 1.63299316185545*x0*x4 - x2*x4 + 0.66666666666667*x0**2
 - 0.81649658092773*x0*x2 + 0.25*x2**2

 + x5**2 - 0.57735026918963*x0*x5 + x1*x5 + 0.08333333333333*x0**2
 - 0.28867513459481*x0*x1 + 0.25*x1**2 - 0.25;

TITLE : 3 lines tangent to four given spheres, each with multiplicity 4

REFERENCES :

Original formulation as polynomial system : Cassiano Durand
(crbd@cs.purdue.edu, http://www.cs.purdue.edu/people/crbd).
Positioning of the centers of the spheres, each with radius
0.5 at the vertices of a tetrahedron : Thorsten Theobald.
(theobald@mathematik.tu-muenchen.de,
 http://www-m9.mathematik.tu-muenchen.de/~theobald/)



Frank Sottile: "From Enumerative Geometry to Solving Systems
  of Polynomial Equations".
In "Computations in Algebraic Geometry with Macaulay 2",
edited by David Eisenbud, Daniel R. Grayson, Michael Stillman,
and Bernd Sturmfels, Volume 8 of Algorithms and Computation
in Mathematics, pages 101-129, Springer-Verlag, 2002.

Frank Sottile and Thorsten Theobald:
"Lines tangents to 2n - 2 spheres in R^n"
Trans. Amer. Math. Soc. 354:4815-4829, 2002

DESCRIPTION :

 c1 = (0, 0, 0)
 c2 = (1, 0, 0);
 c3 = (1/2, sqrt(3)/2, 0)
 c4 = (1/2, sqrt(3)/6, sqrt(6)/3);   

Tangent vector t = (x0,x1,x2) and moment vector m = (x3,x4,x5).
The first equation is ||t||=1, the second m.t = 0,
the other equations are ||m - c_i x t ||^2 - r^2 = 0, where
the radius r = 1/2.

Some constants that are approximated :

sqrt(3)   = 1.73205080756888 
sqrt(3)/2 = 0.86602540378444
sqrt(3)/3 = 0.57735026918963
sqrt(3)/6 = 0.28867513459481 
sqrt(6)/3 = 0.81649658092773
sqrt(6)*2/3 = 1.63299316185545 
sqrt(6)*sqrt(3)/9 = 0.47140452079103
1/12 = 0.08333333333333
2/3  = 0.66666666666667

Because (t,p) and (-t,-p) represent the same line, every line
is represent by two solutions.

THE SOLUTIONS :

24 6
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07106803165780E-01   3.77452918725401E-08
 x1 : -4.08248430737360E-01  -1.83624917064964E-07
 x2 :  5.77350143082334E-01  -8.36140714113780E-08
 x3 : -2.50000000000000E-01  -1.57896818458518E-16
 x4 :  4.33012701892221E-01  -9.11600174682333E-17
 x5 :  9.56878363411174E-08   1.54062878745083E-07
== err :  8.381E-07 = rco :  5.423E-08 = res :  2.562E-13 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07106870026153E-01  -7.33959350196041E-08
 x1 :  4.08248222025870E-01  -2.27096836038030E-08
 x2 : -5.77350208776733E-01   7.38331237214013E-08
 x3 :  2.49999999999994E-01  -3.09771902891427E-16
 x4 : -1.44337506884517E-01   7.38331239002477E-08
 x5 : -4.08248311823044E-01  -2.61039514827208E-08
== err :  5.695E-07 = rco :  2.583E-08 = res :  9.233E-14 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07106794356709E-01  -1.29682370414209E-07
 x1 : -4.08248217029256E-01   1.11010906008961E-07
 x2 :  5.77350304985648E-01  -8.03312536501087E-08
 x3 : -2.50000000000001E-01  -1.74789416181029E-16
 x4 :  4.33012701892220E-01  -1.00914936462574E-16
 x5 : -6.07788020445124E-08  -1.39412292964849E-07
== err :  6.376E-07 = rco :  2.498E-08 = res :  1.416E-13 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07107000502648E-01   2.25297320882032E-09
 x1 : -4.08248198242010E-01   9.32236262450057E-08
 x2 :  5.77350065794056E-01   6.86783756675267E-08
 x3 : -2.50000000000001E-01   3.76366778947993E-16
 x4 :  4.33012701892220E-01   2.17294140272526E-16
 x5 : -1.64133046126442E-07  -6.89421539746136E-08
== err :  9.200E-07 = rco :  4.065E-08 = res :  3.780E-13 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.45837466296011E-09  -7.88468195374747E-08
 x1 : -8.16496606195835E-01  -7.83082011722029E-08
 x2 : -5.77350233455127E-01   1.10744520142767E-07
 x3 : -3.09469859352378E-08  -9.59075677735555E-08
 x4 : -2.88675116727565E-01   5.53722603195699E-08
 x5 :  4.08248303097916E-01   3.91541007615960E-08
== err :  5.650E-07 = rco :  1.596E-08 = res :  8.837E-14 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07106613786070E-01  -1.40554777765690E-07
 x1 :  4.08248469069714E-01  -4.78230374165879E-09
 x2 : -5.77350347919083E-01   1.68762143812694E-07
 x3 :  2.49999999999995E-01   4.27197373015022E-16
 x4 : -1.44337646026867E-01   1.68762143566051E-07
 x5 : -4.08248262628794E-01  -5.96664277999609E-08
== err :  1.071E-06 = rco :  3.356E-08 = res :  3.590E-13 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07106777721982E-01   2.01661826981310E-08
 x1 :  4.08248459401046E-01  -1.48887521817275E-07
 x2 : -5.77350153976196E-01  -1.29977805146137E-07
 x3 :  2.50000000000000E-01  -1.61318992306101E-16
 x4 : -1.44337452083983E-01  -1.29977805052999E-07
 x5 : -4.08248331197955E-01   4.59540935795733E-08
== err :  9.147E-07 = rco :  4.069E-08 = res :  2.574E-13 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -8.32763552714868E-08  -6.75421517130010E-08
 x1 : -8.16496536045323E-01   5.03908816535932E-08
 x2 : -5.77350332662932E-01  -7.12634682544474E-08
 x3 :  5.49694938584535E-08   6.17159738701382E-08
 x4 : -2.88675166331465E-01  -3.56317340426795E-08
 x5 :  4.08248268022662E-01  -2.51954407670146E-08
== err :  6.144E-07 = rco :  1.708E-08 = res :  8.564E-14 ==
solution 9 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -6.76958063497436E-08  -5.52686952253397E-08
 x1 :  8.16496672345763E-01  -8.74590170230574E-08
 x2 :  5.77350139905001E-01   1.23685728025824E-07
 x3 :  1.11963770931823E-07  -1.07114982555938E-07
 x4 :  2.88675069952500E-01   6.18428655357273E-08
 x5 : -4.08248336172882E-01   4.37295095883214E-08
== err :  9.934E-07 = rco :  3.137E-08 = res :  1.586E-13 ==
solution 10 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  3.98191345771439E-08  -7.86209533295894E-08
 x1 :  8.16496542944871E-01   7.46728144685411E-08
 x2 :  5.77350322905497E-01  -1.05603306961978E-07
 x3 : -4.65193071853643E-08   9.14551465527205E-08
 x4 :  2.88675161452747E-01  -5.28016534287562E-08
 x5 : -4.08248271472437E-01  -3.73364071973360E-08
== err :  6.550E-07 = rco :  1.724E-08 = res :  1.033E-13 ==
solution 11 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07106721955267E-01   5.48805917280166E-08
 x1 : -4.08248198711336E-01  -8.25155461304299E-08
 x2 :  5.77350406611668E-01   8.86742103568660E-09
 x3 : -2.50000000000000E-01   6.40569470093182E-16
 x4 :  4.33012701892221E-01   3.69833439654812E-16
 x5 : -4.31664985572901E-08   8.56506523803914E-08
== err :  5.525E-07 = rco :  2.306E-08 = res :  1.179E-13 ==
solution 12 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -7.07106907271332E-01  -5.90137378087707E-09
 x1 :  4.08248253899557E-01   1.86592276258933E-07
 x2 : -5.77350140622791E-01   1.39168341132018E-07
 x3 :  2.49999999999998E-01  -1.26900013858965E-15
 x4 : -1.44337438730577E-01   1.39168341864676E-07
 x5 : -4.08248335919098E-01  -4.92034399066012E-08
== err :  9.849E-07 = rco :  4.011E-08 = res :  2.698E-13 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07106907275902E-01   5.96782319981974E-09
 x1 : -4.08248253845766E-01  -1.86529268655319E-07
 x2 :  5.77350140655230E-01  -1.39205171613295E-07
 x3 : -2.49999999999998E-01   1.29519351582755E-15
 x4 :  1.44337438763016E-01  -1.39205172361075E-07
 x5 :  4.08248335907630E-01   4.92164614695200E-08
== err :  9.848E-07 = rco :  4.010E-08 = res :  2.697E-13 ==
solution 14 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07106721891121E-01  -5.46961908403215E-08
 x1 :  4.08248198454219E-01   8.22199845792108E-08
 x2 : -5.77350406872040E-01  -8.85057057132692E-09
 x3 :  2.50000000000000E-01  -6.08396582858838E-16
 x4 : -4.33012701892221E-01  -3.51258369153850E-16
 x5 :  4.33315599048063E-08  -8.53491333166340E-08
== err :  5.525E-07 = rco :  2.308E-08 = res :  1.177E-13 ==
solution 15 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -3.96329555829235E-08   7.89311242223708E-08
 x1 : -8.16496542971787E-01  -7.47240429846589E-08
 x2 : -5.77350322867433E-01   1.05675755024253E-07
 x3 :  4.64863425693947E-08  -9.15178884151049E-08
 x4 : -2.88675161433715E-01   5.28378774567905E-08
 x5 :  4.08248271485894E-01   3.73620214532009E-08
== err :  6.547E-07 = rco :  1.727E-08 = res :  1.030E-13 ==
solution 16 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  6.76319276665201E-08   5.53146005198789E-08
 x1 : -8.16496672347423E-01   8.74541419584336E-08
 x2 : -5.77350139902654E-01  -1.23678833643316E-07
 x3 : -1.11965803967623E-07   1.07109011845542E-07
 x4 : -2.88675069951327E-01  -6.18394183530339E-08
 x5 :  4.08248336173712E-01  -4.37270720620629E-08
== err :  9.933E-07 = rco :  3.138E-08 = res :  1.584E-13 ==
solution 17 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  8.31847157469980E-08   6.76123701597765E-08
 x1 :  8.16496536044054E-01  -5.03903815566581E-08
 x2 :  5.77350332664727E-01   7.12627610105793E-08
 x3 : -5.49710478312150E-08  -6.17153613789818E-08
 x4 :  2.88675166332362E-01   3.56313804190287E-08
 x5 : -4.08248268022028E-01   2.51951907173332E-08
== err :  6.144E-07 = rco :  1.710E-08 = res :  8.527E-14 ==
solution 18 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07106777739115E-01  -2.01591653362143E-08
 x1 : -4.08248459304272E-01   1.48979944606361E-07
 x2 :  5.77350154023643E-01   1.30034563449017E-07
 x3 : -2.50000000000000E-01   1.59722980477586E-16
 x4 :  1.44337452131430E-01   1.30034563356801E-07
 x5 :  4.08248331181180E-01  -4.59741606713041E-08
== err :  9.146E-07 = rco :  4.067E-08 = res :  2.572E-13 ==
solution 19 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07106613769497E-01   1.40554048614980E-07
 x1 : -4.08248469106515E-01   4.76605364454508E-09
 x2 :  5.77350347913358E-01  -1.68772741342966E-07
 x3 : -2.49999999999995E-01  -4.34818862138328E-16
 x4 :  1.44337646021142E-01  -1.68772741091922E-07
 x5 :  4.08248262630818E-01   5.96701745864973E-08
== err :  1.071E-06 = rco :  3.356E-08 = res :  3.589E-13 ==
solution 20 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.44665910474754E-09   7.89367269521567E-08
 x1 :  8.16496606210238E-01   7.82668051686547E-08
 x2 :  5.77350233434758E-01  -1.10685977353121E-07
 x3 :  3.09646256720724E-08   9.58568682305135E-08
 x4 :  2.88675116717381E-01  -5.53429889312276E-08
 x5 : -4.08248303105118E-01  -3.91334027644044E-08
== err :  5.647E-07 = rco :  1.596E-08 = res :  8.829E-14 ==
solution 21 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07107000675288E-01  -2.28703415682620E-09
 x1 :  4.08248198057130E-01  -9.31261738870242E-08
 x2 : -5.77350065713346E-01  -6.86511824157404E-08
 x3 :  2.50000000000001E-01  -3.48597943633478E-16
 x4 : -4.33012701892220E-01  -2.01261841550752E-16
 x5 :  1.64346461558173E-07   6.88543158603294E-08
== err :  9.197E-07 = rco :  4.067E-08 = res :  3.772E-13 ==
solution 22 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07106794394148E-01   1.29978152423003E-07
 x1 :  4.08248217161770E-01  -1.11083980584439E-07
 x2 : -5.77350304846094E-01   8.06418396205759E-08
 x3 :  2.50000000000001E-01   1.25448976116241E-16
 x4 : -4.33012701892221E-01   7.24280976767629E-17
 x5 :  6.06956282162081E-08   1.39595176303544E-07
== err :  6.365E-07 = rco :  2.502E-08 = res :  1.402E-13 ==
solution 23 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07106869901181E-01   7.34767432925269E-08
 x1 : -4.08248222236349E-01   2.28514782367473E-08
 x2 :  5.77350208780961E-01  -7.38318292927387E-08
 x3 : -2.49999999999994E-01   3.00145404705636E-16
 x4 :  1.44337506888745E-01  -7.38318294660273E-08
 x5 :  4.08248311821550E-01   2.61034938252183E-08
== err :  5.696E-07 = rco :  2.584E-08 = res :  9.219E-14 ==
solution 24 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  7.07106803092791E-01  -3.77695901184261E-08
 x1 :  4.08248430513239E-01   1.83900009240734E-07
 x2 : -5.77350143330205E-01   8.37788318022830E-08
 x3 :  2.50000000000000E-01   1.48462451286493E-16
 x4 : -4.33012701892221E-01   8.57128501166688E-17
 x5 : -9.55513508117388E-08  -1.54279719318310E-07
== err :  8.382E-07 = rco :  5.425E-08 = res :  2.559E-13 ==