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(Implicitization experiments on curves and surfaces)
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{| class="experiments sortable"  
{| class="experiments sortable"  
!  No
!  No
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! curve <ref> Many thanks to Tatjana Kalinka for providing this list of curves and surfaces. </ref>
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! class="unsortable" | curve <ref> Many thanks to Tatjana Kalinka for providing this list of curves and surfaces. </ref>
! class="unsortable" | equation  
! class="unsortable" | equation  
! class="unsortable" | supports
! class="unsortable" | supports
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TOPCOM point2triang(sec) <ref> This is the computation time of '''points2triangs''' client of [http://www.rambau.wm.uni-bayreuth.de/TOPCOM/ TOPCOM] package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.</ref>
TOPCOM point2triang(sec) <ref> This is the computation time of '''points2triangs''' client of [http://www.rambau.wm.uni-bayreuth.de/TOPCOM/ TOPCOM] package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.</ref>
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! <nowiki># mixed cell configurations</nowiki>
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! class="unsortable" | <nowiki># mixed cell configurations</nowiki>
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! <nowiki># extreme terms</nowiki>
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! class="unsortable" | <nowiki># extreme terms</nowiki>
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! <nowiki># all terms</nowiki>
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! class="unsortable" | <nowiki># all terms</nowiki>
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Revision as of 23:35, 26 January 2010

Implicitization experiments on curves and surfaces

No curve [1] equation supports # mixed subdivisions

Enum by reverse search (sec) [2]

TOPCOM point2alltriang (sec) [3]

TOPCOM point2triang(sec) [4]

# mixed cell configurations # extreme terms # all terms
1. astroid Failed to parse (Cannot write to or create math temp directory): a\cos(t)^3,a\sin(t)^3

[curves_supports/supports1.dat supports]

289 193.62 0.048 0.452 289 35 454
2. cardioid Failed to parse (Cannot write to or create math temp directory): a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t))

[curves_supports/supports2.dat supports]

37 6.52 0.005 0.024 37 10 33
3. circle Failed to parse (Cannot write to or create math temp directory): \cos(t),\sin(t)

[curves_supports/supports3.dat supports]

5 0.004 0.016 0.004 5 3 4
4. conchoid Failed to parse (Cannot write to or create math temp directory): a \cos(t),ah \sin(t)

[curves_supports/supports4.dat supports]

12 0.84 0.003 0.008 12 4 6
5. ellipse Failed to parse (Cannot write to or create math temp directory): a\cos(t),b\sin(t)

[curves_supports/supports5.dat supports]

5 0.15 0.001 0.004 5 3 4
6. folium of descartes Failed to parse (Cannot write to or create math temp directory): 3ah/(1+ h^3), 3ah^2/(ah^3)

[curves_supports/supports6.dat supports]

14 0.94 0.004 0.008 14 6 10
7. involute of a circle Failed to parse (Cannot write to or create math temp directory): a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))

[curves_supports/supports7.dat supports]

14 1.00 0.001 0.007 14 6 7
8. nephroid Failed to parse (Cannot write to or create math temp directory): a(3\cos(t)-\cos(3t)),a(3\sin(t)-\sin(3t))

[curves_supports/supports8.dat supports]

289 195.27 0.004 0.240 289 35 454
9a. plateau curve Failed to parse (Cannot write to or create math temp directory): a\sin(3t)/\sin(t),2a\sin(2t)

[curves_supports/supports9a.dat supports]

94 33.02 0.012 0.064 94 15 55
9b. plateau curve Failed to parse (Cannot write to or create math temp directory): a\sin(6t)/ \sin(2t), 2a\sin(4t)

[curves_supports/supports9b.dat supports]

42168 halt 25.934 85.597 42168 495 not computed
10. talbot's curve Failed to parse (Cannot write to or create math temp directory): (a^2 + f^2 \sin( t)^2) \cos( t)/a, (a^2 - 2f^2 + (f^2)\sin(t)^2)\sin(t)/b

[curves_supports/supports10.dat supports]

1944 3948.80 0.416 2.356 1944 84 1600
11. tricuspoid Failed to parse (Cannot write to or create math temp directory): a(2\cos(t)+\cos(2t)),a(2\sin(t)-\sin(2t))

[curves_supports/supports11.dat supports]

37 6.20 0.008 0.024 37 10 33
12. witch of agnesi Failed to parse (Cannot write to or create math temp directory): ah,a/(1 h^2)

[curves_supports/supports12.dat supports]

2 0.03 0.007 0.004 2 2 2
13. circle (3 systems) Failed to parse (Cannot write to or create math temp directory): (-t^2 +1)/s, 2t/s, t^2 -s +1

[curves_supports/supports13.dat supports]

26 6.00 0.020 0.052 26 6 7


No surface equation supports # mixed subdivisions

Enum by reverse search (sec)

TOPCOM point2alltriang (sec)

TOPCOM point2triang(sec)

# mixed cell configurations # extreme terms # all terms
1. cylinder Failed to parse (Cannot write to or create math temp directory): \cos(t),\sin(t),s

[surfaces_supports/supports1.dat supports]

5 0.24 0.003 0.006 5 3 4
2. cone Failed to parse (Cannot write to or create math temp directory): s\cos(t),s\sin(t),s

[surfaces_supports/supports2.dat supports]

122 73.45 0.192 0.288 98 8 14
3. paraboloid Failed to parse (Cannot write to or create math temp directory): s\cos(t),s\sin(t),s^2

[surfaces_supports/supports3.dat supports]

122 71.60 0.192 0.296 98 8 37
4. surface of revolution Failed to parse (Cannot write to or create math temp directory): s\cos(t),s\sin(t),\cos(s)

[surfaces_supports/supports4.dat supports]

122 71.80 0.193 0.288 98 8 37
5. sphere Failed to parse (Cannot write to or create math temp directory): \sin(t)\cos(s),\sin(t)\sin(s),\cos(t)

[surfaces_supports/supports5.dat supports]

104148 halt 19496.602 714.161 43018 21 186
6. sphere2 Failed to parse (Cannot write to or create math temp directory): \cos(t)\cos(s),\sin(t)\cos(s),\sin(s)

[surfaces_supports/supports6.dat supports]

76280 halt 4492.977 397.157 32076 95 776
7. stereographic shpere Failed to parse (Cannot write to or create math temp directory): 2t/(1 t^2 s^2),2s/(1 t^2 s^2),(t^2 s^2-1)/(1 t^2 s^2)

[surfaces_supports/supports7.dat supports]

3540 7112.54 25.402 11.025 3126 22 283
8. twisted shpere Failed to parse (Cannot write to or create math temp directory): a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))

[surfaces_supports/supports8.dat supports]

>1812221 not computed not computed not computed not computed not computed not computed

Remarks

  1. Many thanks to Tatjana Kalinka for providing this list of curves and surfaces.
  2. This is the computation time of enumeration of regular triangulations algorithm using reverse search. I would like to thank very much Fumihiko TAKEUCHI for running the experiments and providing this results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.
  3. This is the computation time of points2alltriangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.
  4. This is the computation time of points2triangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.
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