6
 x0**2 + x1**2 + x2**2 - 1;
 x0*x3 + x1*x4 + x2*x5;
 x3**2 + x4**2 + x5**2 - 0.3;
 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 : 12 real lines tangent to four given spheres.

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 at the vertices of a
tetrahedron, with suggestion to enlarge radii : 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.
The radius of the first sphere is enlarged so that r^2 = 0.3.
The radii of the other spheres are 0.5.

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, there are 12
tangent lines.

THE SOLUTIONS :

24 6
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -1.43102286084844E-01   1.79276319039945E-62
 x1 : -6.40171251688474E-01  -1.79276319039945E-62
 x2 : -7.54786396425442E-01   1.33697593860298E-62
 x3 :  1.69934603999895E-01  -1.42813338896227E-62
 x4 : -4.12550523019043E-01   1.06350358752510E-62
 x5 :  3.17685845325398E-01   1.33697593860298E-62
== err :  1.547E-15 = rco :  2.573E-02 = res :  1.704E-16 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -6.25955709777127E-01   1.23464623112765E-57
 x1 :  1.96155410755113E-01   1.75240110224570E-57
 x2 : -7.54786396425443E-01  -7.56718657787915E-58
 x3 :  2.72311931279100E-01  -1.99136488891557E-58
 x4 : -3.53442945555486E-01  -3.98272977783113E-58
 x5 : -3.17685845325391E-01  -3.98272977783113E-58
== err :  1.334E-15 = rco :  6.301E-02 = res :  2.290E-16 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -3.80665581868995E-01  -4.66726145839586E-61
 x1 :  5.11993743627026E-01  -7.77876909732643E-61
 x2 : -7.70036441518921E-01   4.86173068582902E-61
 x3 :  2.27441105124784E-01   5.83407682299482E-62
 x4 : -3.55691615760317E-01   6.80642296016062E-61
 x5 : -3.48932684306632E-01  -3.50044609379689E-61
== err :  1.923E-15 = rco :  2.817E-02 = res :  4.094E-16 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -6.71265049771219E-01   2.58279442684667E-63
 x1 :  6.79771424646732E-01   3.95015618223608E-63
 x2 : -2.95489497595760E-01   1.06350358752510E-63
 x3 :  2.73861116201760E-01   7.14040156441371E-64
 x4 :  6.62646433320782E-02  -3.79822709830392E-65
 x5 : -4.69690415142569E-01   3.98813845321912E-64
== err :  1.350E-15 = rco :  3.103E-02 = res :  2.429E-16 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -3.80665581869013E-01   5.22839756023991E-62
 x1 : -5.55332738202709E-01   1.21543267145725E-63
 x2 :  7.39391144564639E-01   3.52475474722604E-62
 x3 : -2.27441105124789E-01   8.28013507430254E-63
 x4 :  4.47540761587898E-01  -9.87539045559019E-64
 x5 :  2.19038376584879E-01   1.29139721342333E-62
== err :  1.380E-15 = rco :  3.043E-02 = res :  1.596E-16 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -8.72633177299453E-01  -5.74226793183870E-53
 x1 : -2.18976395263573E-01   2.08809742975953E-53
 x2 :  4.36532560290333E-01   1.14845358636774E-52
 x3 : -2.25263630757588E-01  -3.65417050207917E-53
 x4 :  4.44494751511597E-01   2.61012178719941E-53
 x5 : -2.27333922977968E-01  -5.22024357439882E-53
== err :  2.148E-15 = rco :  2.780E-02 = res :  3.747E-16 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -6.71265049771223E-01  -1.05590713332849E-62
 x1 : -5.19996951435873E-02  -2.70433769399239E-62
 x2 :  7.39391144564642E-01  -1.21543267145725E-62
 x3 : -2.73861116201760E-01   2.50682988488059E-63
 x4 :  4.20740155696822E-01   5.16558885369333E-63
 x5 : -2.19038376584882E-01   5.61698206620287E-63
== err :  1.663E-15 = rco :  2.706E-02 = res :  3.123E-16 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -1.43102286084821E-01   2.61128113008394E-64
 x1 : -9.25009856242540E-01   4.67359974986615E-66
 x2 : -3.51963778777665E-01  -1.18694596821997E-65
 x3 : -1.69934603999891E-01   1.54302975868597E-65
 x4 : -1.62000246349090E-01  -2.72997572690594E-65
 x5 :  4.94851644986893E-01  -1.57270340789147E-65
== err :  4.420E-15 = rco :  2.810E-02 = res :  4.337E-16 ==
solution 9 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -8.72633177299466E-01   1.70160574004016E-62
 x1 :  3.38574713032152E-01   3.00819586185670E-62
 x2 : -3.51963778777639E-01  -9.95135499755627E-63
 x3 :  2.25263630757580E-01   7.86233009348911E-63
 x4 :  6.61675608714289E-02  -9.83740818460715E-63
 x5 : -4.94851644986898E-01   2.35490080094843E-63
== err :  1.559E-15 = rco :  2.743E-02 = res :  1.596E-16 ==
solution 10 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -2.53066797625170E-01   2.12700717505019E-62
 x1 :  9.21218298097868E-01   1.29139721342333E-63
 x2 :  2.95489497595785E-01   9.72346137165803E-63
 x3 :  1.94317422599166E-01  -2.43086534291451E-63
 x4 :  2.04038362073452E-01   2.46884761389755E-63
 x5 : -4.69690415142570E-01   5.50742929254068E-64
== err :  2.351E-15 = rco :  2.633E-02 = res :  1.908E-16 ==
solution 11 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -6.25955709777125E-01   0.00000000000000E+00
 x1 : -6.46234302094660E-01  -4.27300548559191E-64
 x2 :  4.36532560290345E-01  -5.69734064745588E-64
 x3 : -2.72311931279100E-01  -2.84867032372794E-65
 x4 :  4.17331402540592E-01  -1.70920219423676E-64
 x5 :  2.27333922977959E-01   3.94066061449032E-64
== err :  1.636E-15 = rco :  3.234E-02 = res :  1.596E-16 ==
solution 12 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 : -2.53066797625191E-01  -2.73472351077882E-63
 x1 :  5.85662936058455E-01  -8.73592232609901E-64
 x2 :  7.70036441518920E-01   5.31751793762549E-64
 x3 : -1.94317422599172E-01  -2.65875896881274E-64
 x4 :  3.74815582783031E-01   6.07716335728627E-64
 x5 : -3.48932684306632E-01   3.41840438847353E-64
== err :  2.018E-15 = rco :  2.710E-02 = res :  1.110E-16 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  2.53066797625191E-01  -5.92523427335411E-63
 x1 : -5.85662936058455E-01  -3.07656394962617E-63
 x2 : -7.70036441518920E-01   9.49556774575980E-64
 x3 :  1.94317422599172E-01  -9.80417369749699E-64
 x4 : -3.74815582783031E-01  -1.89911354915196E-65
 x5 :  3.48932684306632E-01   6.17211903474387E-64
== err :  1.483E-15 = rco :  2.710E-02 = res :  1.110E-16 ==
solution 14 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  6.25955709777125E-01  -4.08309413067671E-64
 x1 :  6.46234302094660E-01   4.93769522779510E-64
 x2 : -4.36532560290345E-01   3.41840438847353E-64
 x3 :  2.72311931279100E-01  -4.74778387287990E-66
 x4 : -4.17331402540592E-01   2.84867032372794E-64
 x5 : -2.27333922977959E-01  -4.74778387287990E-64
== err :  1.636E-15 = rco :  3.234E-02 = res :  1.596E-16 ==
solution 15 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  2.53066797625170E-01  -1.15466103788439E-62
 x1 : -9.21218298097868E-01  -9.11574503592941E-64
 x2 : -2.95489497595785E-01  -4.25401435010039E-63
 x3 : -1.94317422599166E-01  -1.51929083932157E-64
 x4 : -2.04038362073452E-01  -3.00059940766010E-63
 x5 :  4.69690415142570E-01  -3.01959054315162E-63
== err :  2.351E-15 = rco :  2.633E-02 = res :  1.908E-16 ==
solution 16 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  8.72633177299466E-01   2.84867032372794E-64
 x1 : -3.38574713032152E-01   6.07716335728627E-64
 x2 :  3.51963778777639E-01   1.29139721342333E-63
 x3 : -2.25263630757580E-01  -1.06350358752510E-63
 x4 : -6.61675608714289E-02   8.35609961626862E-64
 x5 :  4.94851644986898E-01  -2.39058931669106E-64
== err :  1.559E-15 = rco :  2.743E-02 = res :  1.596E-16 ==
solution 17 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  1.43102286084821E-01  -1.19881542790217E-64
 x1 :  9.25009856242540E-01   5.93472984109987E-67
 x2 :  3.51963778777665E-01   5.93472984109987E-66
 x3 :  1.69934603999891E-01  -4.30267913479741E-66
 x4 :  1.62000246349090E-01   1.51335610948047E-65
 x5 : -4.94851644986893E-01   6.89912344027860E-66
== err :  4.420E-15 = rco :  2.810E-02 = res :  4.337E-16 ==
solution 18 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  6.71265049771223E-01   1.82314900718588E-62
 x1 :  5.19996951435873E-02   3.72545613959992E-62
 x2 : -7.39391144564642E-01   1.70160574004016E-62
 x3 :  2.73861116201760E-01  -4.02612072420215E-63
 x4 : -4.20740155696822E-01  -7.44452511267568E-63
 x5 :  2.19038376584882E-01  -7.14066694481137E-63
== err :  1.663E-15 = rco :  2.706E-02 = res :  3.123E-16 ==
solution 19 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  8.72633177299453E-01   3.65417050207917E-52
 x1 :  2.18976395263573E-01   3.96738511654310E-52
 x2 : -4.36532560290333E-01   4.38500460249501E-52
 x3 :  2.25263630757588E-01   1.07014993275176E-52
 x4 : -4.44494751511597E-01  -6.75369012437847E-53
 x5 :  2.27333922977968E-01  -3.65417050207917E-53
== err :  5.548E-15 = rco :  2.780E-02 = res :  2.151E-16 ==
solution 20 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  3.80665581869013E-01  -8.14339889876360E-62
 x1 :  5.55332738202709E-01  -1.45851920574871E-62
 x2 : -7.39391144564640E-01  -6.80642296016062E-62
 x3 :  2.27441105124789E-01  -1.85353482397231E-62
 x4 : -4.47540761587898E-01  -9.72346137165803E-63
 x5 : -2.19038376584879E-01  -3.03858167864314E-63
== err :  1.516E-15 = rco :  3.043E-02 = res :  2.498E-16 ==
solution 21 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  6.71265049771219E-01  -4.25401435010039E-63
 x1 : -6.79771424646732E-01  -6.68487969301490E-63
 x2 :  2.95489497595760E-01  -3.34243984650745E-63
 x3 : -2.73861116201760E-01  -9.49556774575980E-64
 x4 : -6.62646433320782E-02  -7.59645419660784E-64
 x5 :  4.69690415142569E-01  -7.97627690643823E-64
== err :  1.350E-15 = rco :  3.103E-02 = res :  2.429E-16 ==
solution 22 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  3.80665581868995E-01   8.75111523449223E-62
 x1 : -5.11993743627026E-01   1.06958075088238E-61
 x2 :  7.70036441518921E-01  -4.61864415153757E-62
 x3 : -2.27441105124784E-01  -1.70160574004016E-62
 x4 :  3.55691615760317E-01  -6.56333642586917E-62
 x5 :  3.48932684306632E-01   3.70706964794463E-62
== err :  1.923E-15 = rco :  2.817E-02 = res :  4.094E-16 ==
solution 23 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  6.25955709777127E-01   6.37236764452981E-58
 x1 : -1.96155410755113E-01  -1.21074985246066E-56
 x2 :  7.54786396425443E-01  -3.14635652448659E-57
 x3 : -2.72311931279100E-01   1.61300556002161E-57
 x4 :  3.53442945555486E-01  -4.73944843561905E-57
 x5 :  3.17685845325391E-01   6.05374926230332E-57
== err :  1.334E-15 = rco :  6.301E-02 = res :  2.290E-16 ==
solution 24 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x0 :  1.43102286084844E-01   2.43086534291451E-63
 x1 :  6.40171251688474E-01  -2.35490080094843E-63
 x2 :  7.54786396425442E-01   3.89318277576152E-64
 x3 : -1.69934603999895E-01  -4.38318463518646E-64
 x4 :  4.12550523019043E-01   1.40534402637245E-63
 x5 : -3.17685845325398E-01   3.03858167864314E-63
== err :  1.547E-15 = rco :  2.573E-02 = res :  1.704E-16 ==