6
- d1*q1 - d2*q2 - d3*q3 + 1;
- 3.6*d1*q1 + 4.1*d1*q2 + 2.0*d1*q3 + 0.1*d1 + 4.1*d2*q1 + 1.8*d2*q2
+ 3.7*d2*q3 - 0.2*d2 + 2.0*d3*q1 + 3.7*d3*q2 - 4.0*d3*q3 + 0.3*d3 +
0.1*q1 - 0.2*q2 + 0.3*q3 + 5.8;
- 2.140796*d1*q1 - 3.998792*d1*q2 + 3.715992*d1*q3 - 0.2828*d1 -
3.998792*d2*q1 - 1.575196*d2*q2 - 3.998792*d2*q3 + 3.715992*d3*q1 -
3.998792*d3*q2 - 2.140796*d3*q3 + 0.2828*d3 - 0.2828*q1 + 0.2828*q3 + 5.856788;
0.3464*d1*q1 + 0.1732*d1*q2 - 5.999648*d1*q3 - 0.1732*d1 + 0.1732*d2*
q1 - 5.999648*d2*q2 - 0.1732*d2*q3 + 0.3464*d2 - 5.999648*d3*q1 -
0.1732*d3*q2 - 0.3464*d3*q3 - 0.1732*d3 - 0.1732*q1 + 0.3464*q2 -
0.1732*q3 + 5.999648;
- 5701.3*d1*q1 - 2.9*d1*q2 + 3796.7*d1*q3 - 1902.7*d1 - 2.9*d2*q1 -
5698.7*d2*q2 + 1897.3*d2*q3 + 3803.3*d2 + 3796.7*d3*q1 + 1897.3*d3*q2
+ 5703.1*d3*q3 + 0.7*d3 - 1902.7*q1 + 3803.3*q2 + 0.7*q3 + 5696.9;
- 6.8*d1*q1 - 3.2*d1*q2 + 1.3*d1*q3 + 5.1*d1 - 3.2*d2*q1 - 4.8*d2*q2
- 0.7*d2*q3 - 7.1*d2 + 1.3*d3*q1 - 0.7*d3*q2 + 9.0*d3*q3 - d3 + 5.1*q1
- 7.1*q2 - q3 + 2.6;
TITLE : camera displacement between two positions, scaled first frame.
ROOT COUNTS :
total degree : 64
2-homogeneous Bezout bound : 20
with partition : {{d1 d2 d3 }{q1 q2 q3 }}
mixed volume : 20
REFERENCES :
Ioannis Z. Emiris:
`Sparse Elimination and Application in Kinematics'
PhD Thesis, Computer Science, University of California at Berkeley, 1994.
Ioannis Z. Emiris:
`A general Solver Based on Sparse Resultants:
Numerical Issues and Kinematic Applications',
INRIA Rapport de Recherche no 3110, January 1997, 29 pages
Available via anonymous ftp to ftp.inria.fr
NOTE :
This system models the displacement of a camera between two positions
in a static environment, coordinates of matched points in first instance.
The coordinates of the frames have been scaled, i.e., all components have
been divided by 1000.
THE SOLUTIONS :
20 6
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -1.00187746343441E+01 -1.02003169143904E-67
q1 : 2.16349900882507E+00 -2.81088278606781E-68
d2 : 5.27096688651916E+00 4.40468230394131E-68
q2 : 4.14807645389239E+00 -5.44789653382215E-68
d3 : 3.46044449033976E-01 3.18759903574700E-69
q3 : 2.34430970383515E+00 -3.44840259321721E-68
== err : 3.403E-14 = rco : 1.837E-06 = res : 2.183E-11 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -4.00074964087066E-01 1.86690458335834E-60
q1 : 7.14374212666202E+01 -2.54894705781192E-57
d2 : 4.15805732418402E-01 -7.46761833343337E-60
q2 : 6.27398271488606E+01 -2.30998327114206E-57
d3 : -3.88155169000059E-01 5.60071375007503E-60
q3 : -8.99831883542215E+00 2.04114901113845E-58
== err : 1.114E-13 = rco : 4.472E-08 = res : 2.910E-11 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -4.11362919961996E-03 6.34702729602409E-01
q1 : 1.07073370333179E+00 -5.28986736899611E+00
d2 : 2.97394007311655E-01 -3.71117214378002E+00
q2 : 7.08792243065625E-02 1.94181773315705E-01
d3 : 2.72180047596179E-02 5.59540380075883E-01
q3 : -1.15287061420216E+00 5.47490578453015E+00
== err : 2.672E-15 = rco : 5.284E-06 = res : 3.750E-12 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 2.39689707893600E-02 2.03752779273601E-71
q1 : -1.85992426440753E+02 -3.33828553561868E-67
d2 : -1.86554926678331E-02 -2.32493948224955E-71
q2 : -1.07460371835044E+02 -2.22552369041245E-67
d3 : -2.54759702551365E-02 -4.75423151638402E-71
q3 : -1.35552083903577E+02 -1.94733322911090E-67
== err : 2.907E-13 = rco : 2.823E-09 = res : 1.819E-11 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -2.56263047259548E+00 1.00151652128078E-60
q1 : 8.11735863421135E+00 2.17805534725140E-60
d2 : 1.35645958316042E+00 -1.25918824762972E-60
q2 : 1.57220206476798E+01 4.04495993060974E-60
d3 : 5.38269606136141E-02 -4.80095905225616E-62
q3 : 8.83395631861776E+00 2.17805534725140E-60
== err : 4.462E-14 = rco : 4.721E-07 = res : 2.842E-14 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -4.11362919961983E-03 -6.34702729602409E-01
q1 : 1.07073370333179E+00 5.28986736899611E+00
d2 : 2.97394007311654E-01 3.71117214378002E+00
q2 : 7.08792243065624E-02 -1.94181773315705E-01
d3 : 2.72180047596180E-02 -5.59540380075882E-01
q3 : -1.15287061420216E+00 -5.47490578453015E+00
== err : 3.759E-15 = rco : 5.284E-06 = res : 4.547E-12 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 4.31525045865282E-03 4.59858703221669E-73
q1 : 1.53964291762927E+01 1.26779507103574E-70
d2 : 1.57557671967738E-01 -4.10335458259335E-72
q2 : 1.19603230035780E+01 8.15011117094404E-71
d3 : 3.44291835195695E-02 0.00000000000000E+00
q3 : -2.76184329480271E+01 0.00000000000000E+00
== err : 3.066E-14 = rco : 3.142E-07 = res : 4.775E-12 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 2.54964137276597E+01 -1.44890865261227E-70
q1 : 3.54809887556081E-01 -2.26391976970668E-72
d2 : 1.30967935950199E+01 -5.43340744729603E-71
q2 : -3.56996638737865E-01 -1.13195988485334E-72
d3 : -9.06123055507048E+00 9.05567907882671E-71
q3 : 3.72010001913294E-01 4.24484956820002E-72
== err : 1.310E-14 = rco : 3.754E-07 = res : 1.455E-11 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 1.07073370333179E+00 -5.28986736899611E+00
q1 : -4.11362919961994E-03 6.34702729602409E-01
d2 : 7.08792243065625E-02 1.94181773315705E-01
q2 : 2.97394007311655E-01 -3.71117214378002E+00
d3 : -1.15287061420216E+00 5.47490578453015E+00
q3 : 2.72180047596179E-02 5.59540380075883E-01
== err : 4.664E-15 = rco : 4.608E-06 = res : 1.095E-11 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 7.14374212666199E+01 -4.35611069450280E-59
q1 : -4.00074964087066E-01 -9.72346137165803E-63
d2 : 6.27398271488603E+01 -3.98272977783113E-59
q2 : 4.15805732418403E-01 1.40990189889042E-61
d3 : -8.99831883542212E+00 4.01862763508772E-60
q3 : -3.88155169000059E-01 -8.75111523449223E-62
== err : 1.433E-13 = rco : 4.472E-08 = res : 1.455E-11 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -5.44504357673233E-02 -9.33452291679171E-61
q1 : -2.53902514663185E+01 8.96114200012004E-59
d2 : -1.30887818487475E-02 -2.91703841149741E-62
q2 : -1.35197609918457E+01 5.60071375007503E-59
d3 : 1.18597460143422E-02 1.26404997831554E-61
q3 : -4.71736458922360E+01 -1.19481893334934E-58
== err : 3.343E-14 = rco : 1.800E-07 = res : 5.230E-12 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 1.52525122261781E+01 1.03234741498625E-69
q1 : 2.94025403132356E+00 -2.51861074379868E-71
d2 : -1.59650216570430E+01 -1.08668148945921E-69
q2 : 2.07407537095237E+00 -2.00922879561468E-71
d3 : 1.51939278512973E+01 1.03234741498625E-69
q3 : -7.06440260233406E-01 -3.50907564304535E-71
== err : 1.971E-13 = rco : 4.927E-07 = res : 5.821E-11 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 2.16349900882506E+00 -3.41840438847353E-64
q1 : -1.00187746343442E+01 -1.51929083932157E-63
d2 : 4.14807645389238E+00 -7.21663148677745E-64
q2 : 5.27096688651919E+00 3.41840438847353E-64
d3 : 2.34430970383515E+00 -1.70920219423676E-64
q3 : 3.46044449033977E-01 3.32344871101593E-65
== err : 9.576E-15 = rco : 1.830E-06 = res : 1.455E-11 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 2.94025403132356E+00 -1.11276184520623E-67
q1 : 1.52525122261781E+01 -3.17137125883775E-66
d2 : 2.07407537095237E+00 -4.40468230394131E-68
q2 : -1.59650216570430E+01 2.96736492054994E-66
d3 : -7.06440260233406E-01 -9.27301537671855E-69
q3 : 1.51939278512973E+01 -3.26410141260493E-66
== err : 2.443E-14 = rco : 4.927E-07 = res : 4.366E-11 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -1.85992426440753E+02 4.07831529249908E-56
q1 : 2.39689707893600E-02 1.92913473613695E-59
d2 : -1.07460371835044E+02 2.03915764624954E-56
q2 : -1.86554926678331E-02 8.71222138900560E-60
d3 : -1.35552083903577E+02 1.01957882312477E-56
q3 : -2.54759702551365E-02 -8.82890292546549E-60
== err : 2.534E-13 = rco : 2.823E-09 = res : 2.274E-13 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 1.07073370333179E+00 5.28986736899611E+00
q1 : -4.11362919961979E-03 -6.34702729602409E-01
d2 : 7.08792243065624E-02 -1.94181773315705E-01
q2 : 2.97394007311654E-01 3.71117214378002E+00
d3 : -1.15287061420216E+00 -5.47490578453015E+00
q3 : 2.72180047596180E-02 -5.59540380075883E-01
== err : 4.058E-15 = rco : 4.608E-06 = res : 1.819E-12 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 3.54809887556081E-01 -5.11261178461591E-75
q1 : 2.54964137276597E+01 3.53737464016668E-74
d2 : -3.56996638737865E-01 2.76357393763022E-75
q2 : 1.30967935950199E+01 1.76868732008334E-74
d3 : 3.72010001913294E-01 -5.52714787526044E-76
q3 : -9.06123055507050E+00 -1.06121239205001E-73
== err : 4.185E-14 = rco : 3.753E-07 = res : 7.276E-12 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : -2.53902514663185E+01 5.11150998216783E-64
q1 : -5.44504357673233E-02 -2.04006338287808E-67
d2 : -1.35197609918457E+01 3.03858167864314E-64
q2 : -1.30887818487475E-02 0.00000000000000E+00
d3 : -4.71736458922361E+01 6.07716335728627E-64
q3 : 1.18597460143422E-02 -2.07715544438496E-66
== err : 4.941E-14 = rco : 1.800E-07 = res : 7.276E-12 ==
solution 19 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 8.11735863421133E+00 -4.98062349335469E-71
q1 : -2.56263047259548E+00 -1.58474383879467E-71
d2 : 1.57220206476798E+01 1.81113581576534E-70
q2 : 1.35645958316043E+00 -1.01876389636801E-71
d3 : 8.83395631861774E+00 -4.52783953941336E-71
q3 : 5.38269606136141E-02 -3.11288968334668E-72
== err : 2.370E-14 = rco : 4.722E-07 = res : 2.910E-11 ==
solution 20 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
d1 : 1.53964291762927E+01 4.40468230394131E-68
q1 : 4.31525045865282E-03 -8.15011117094404E-71
d2 : 1.19603230035780E+01 2.46314470944087E-68
q2 : 1.57557671967738E-01 -3.62227163153069E-71
d3 : -2.76184329480271E+01 1.85460307534371E-68
q3 : 3.44291835195695E-02 -1.44890865261227E-70
== err : 4.201E-14 = rco : 3.241E-07 = res : 2.842E-14 ==