Implicitization
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(→Implicitization experiments on curves and surfaces) |
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| 1. | | 1. | ||
|| astroid | || astroid | ||
| - | || | + | ||a\cos(t)^3,a\sin(t)^3 |
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[http://ergawiki.di.uoa.gr/curves_supports/supports1.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports1.dat supports] | ||
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| 2. | | 2. | ||
| cardioid | | cardioid | ||
| - | | | + | | $a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t))$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports2.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports2.dat supports] | ||
| Line 52: | Line 52: | ||
| 3. | | 3. | ||
| circle | | circle | ||
| - | | | + | |$ \cos(t),\sin(t)$ |
| | | | ||
[http://ergawiki.di.uoa.gr/curves_supports/supports3.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports3.dat supports] | ||
| Line 66: | Line 66: | ||
| 4. | | 4. | ||
| conchoid | | conchoid | ||
| - | | | + | | $a \cos(t),ah \sin(t)$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports4.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports4.dat supports] | ||
| Line 80: | Line 80: | ||
| 5. | | 5. | ||
| ellipse | | ellipse | ||
| - | | | + | | $a\cos(t),b\sin(t)$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports5.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports5.dat supports] | ||
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| 6. | | 6. | ||
| folium of descartes | | folium of descartes | ||
| - | | | + | | $3ah/(1+ h^3), 3ah^2/(ah^3)$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports6.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports6.dat supports] | ||
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| 7. | | 7. | ||
| involute of a circle | | involute of a circle | ||
| - | | | + | | $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports7.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports7.dat supports] | ||
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| 8. | | 8. | ||
| nephroid | | nephroid | ||
| - | | | + | | $a(3\cos(t)-\cos(3t)),a(3\sin(t)-\sin(3t))$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports8.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports8.dat supports] | ||
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| 9a. | | 9a. | ||
| plateau curve | | plateau curve | ||
| - | | | + | | $a\sin(3t)/\sin(t),2a\sin(2t)$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports9a.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports9a.dat supports] | ||
| Line 150: | Line 150: | ||
| 9b. | | 9b. | ||
| plateau curve | | plateau curve | ||
| - | | | + | | $a\sin(6t)/ \sin(2t), 2a\sin(4t)$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports9b.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports9b.dat supports] | ||
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| 10. | | 10. | ||
| talbot's curve | | talbot's curve | ||
| - | | | + | | $(a^2 + f^2 \sin( t)^2) \cos( t)/a, (a^2 - 2f^2 + (f^2)\sin(t)^2)\sin(t)/b $ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports10.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports10.dat supports] | ||
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| 11. | | 11. | ||
| tricuspoid | | tricuspoid | ||
| - | | | + | | $a(2\cos(t)+\cos(2t)),a(2\sin(t)-\sin(2t))$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports11.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports11.dat supports] | ||
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| 12. | | 12. | ||
| witch of agnesi | | witch of agnesi | ||
| - | | | + | | $ah,a/(1 h^2)$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports12.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports12.dat supports] | ||
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| 13. | | 13. | ||
| circle (3 systems) | | circle (3 systems) | ||
| - | | | + | | $(-t^2 +1)/s, 2t/s, t^2 -s +1$ |
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[http://ergawiki.di.uoa.gr/curves_supports/supports13.dat supports] | [http://ergawiki.di.uoa.gr/curves_supports/supports13.dat supports] | ||
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| 1. | | 1. | ||
| cylinder | | cylinder | ||
| - | | | + | | $\cos(t),\sin(t),s$ |
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[http://ergawiki.di.uoa.gr/surfaces_supports/supports1.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports1.dat supports] | ||
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| 2. | | 2. | ||
| cone | | cone | ||
| - | | | + | | $s\cos(t),s\sin(t),s$ |
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[http://ergawiki.di.uoa.gr/surfaces_supports/supports2.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports2.dat supports] | ||
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| 3. | | 3. | ||
| paraboloid | | paraboloid | ||
| - | | | + | | $s\cos(t),s\sin(t),s^2$ |
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[http://ergawiki.di.uoa.gr/surfaces_supports/supports3.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports3.dat supports] | ||
| Line 283: | Line 283: | ||
| 4. | | 4. | ||
| surface of revolution | | surface of revolution | ||
| - | | | + | | $s\cos(t),s\sin(t),\cos(s)$ |
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[http://ergawiki.di.uoa.gr/surfaces_supports/supports4.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports4.dat supports] | ||
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| 5. | | 5. | ||
| sphere | | sphere | ||
| - | | | + | | $\sin(t)\cos(s),\sin(t)\sin(s),\cos(t)$ |
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[http://ergawiki.di.uoa.gr/surfaces_supports/supports5.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports5.dat supports] | ||
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| 6. | | 6. | ||
| sphere2 | | sphere2 | ||
| - | | | + | | $\cos(t)\cos(s),\sin(t)\cos(s),\sin(s)$ |
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[http://ergawiki.di.uoa.gr/surfaces_supports/supports6.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports6.dat supports] | ||
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| 7. | | 7. | ||
| stereographic shpere | | stereographic shpere | ||
| - | | | + | | $2t/(1 t^2 s^2),2s/(1 t^2 s^2),(t^2 s^2-1)/(1 t^2 s^2)$ |
| | | | ||
[http://ergawiki.di.uoa.gr/surfaces_supports/supports7.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports7.dat supports] | ||
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| 8. | | 8. | ||
| twisted shpere | | twisted shpere | ||
| - | | | + | | $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$ |
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[http://ergawiki.di.uoa.gr/surfaces_supports/supports8.dat supports] | [http://ergawiki.di.uoa.gr/surfaces_supports/supports8.dat supports] | ||
Revision as of 15:02, 23 February 2010
Contents |
Implicitization experiments on curves and surfaces
Curves
| No | curve [1] | equation | supports | # mixed subdivisions |
Enum by reverse search (sec) [2] |
TOPCOM point2alltriang (sec) [3] |
TOPCOM point2triang(sec) [4] | # mixed cell configurations | # N(R) extreme terms | N(R) extreme terms | # all terms |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1. | astroid | a\cos(t)^3,a\sin(t)^3 | 289 | 193.62 | 0.048 | 0.452 | 289 | 35 | N(R) | 454 | |
| 2. | cardioid | $a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t))$ | 37 | 6.52 | 0.005 | 0.024 | 37 | 10 | N(R) | 33 | |
| 3. | circle | $ \cos(t),\sin(t)$ | 5 | 0.004 | 0.016 | 0.004 | 5 | 3 | N(R) | 4 | |
| 4. | conchoid | $a \cos(t),ah \sin(t)$ | 12 | 0.84 | 0.003 | 0.008 | 12 | 4 | N(R) | 6 | |
| 5. | ellipse | $a\cos(t),b\sin(t)$ | 5 | 0.15 | 0.001 | 0.004 | 5 | 3 | N(R) | 4 | |
| 6. | folium of descartes | $3ah/(1+ h^3), 3ah^2/(ah^3)$ | 14 | 0.94 | 0.004 | 0.008 | 14 | 6 | N(R) | 10 | |
| 7. | involute of a circle | $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$ | 14 | 1.00 | 0.001 | 0.007 | 14 | 6 | N(R) | 7 | |
| 8. | nephroid | $a(3\cos(t)-\cos(3t)),a(3\sin(t)-\sin(3t))$ | 289 | 195.27 | 0.004 | 0.240 | 289 | 35 | N(R) | 454 | |
| 9a. | plateau curve | $a\sin(3t)/\sin(t),2a\sin(2t)$ | 94 | 33.02 | 0.012 | 0.064 | 94 | 15 | N(R) | 55 | |
| 9b. | plateau curve | $a\sin(6t)/ \sin(2t), 2a\sin(4t)$ | 42168 | halt | 25.934 | 85.597 | 42168 | 495 | N(R) | not computed | |
| 10. | talbot's curve | $(a^2 + f^2 \sin( t)^2) \cos( t)/a, (a^2 - 2f^2 + (f^2)\sin(t)^2)\sin(t)/b $ | 1944 | 3948.80 | 0.416 | 2.356 | 1944 | 84 | N(R) | 1600 | |
| 11. | tricuspoid | $a(2\cos(t)+\cos(2t)),a(2\sin(t)-\sin(2t))$ | 37 | 6.20 | 0.008 | 0.024 | 37 | 10 | N(R) | 33 | |
| 12. | witch of agnesi | $ah,a/(1 h^2)$ | 2 | 0.03 | 0.007 | 0.004 | 2 | 2 | N(R) | 2 | |
| 13. | circle (3 systems) | $(-t^2 +1)/s, 2t/s, t^2 -s +1$ | 26 | 6.00 | 0.020 | 0.052 | 26 | 6 | N(R) | 7 |
Surfaces
| No | surface | equation | supports | # mixed subdivisions |
Enum by reverse search (sec) |
TOPCOM point2alltriang (sec) |
TOPCOM point2triang(sec) | # mixed cell configurations | # N(R) extreme terms | N(R) extreme terms | # all terms |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1. | cylinder | $\cos(t),\sin(t),s$ | 5 | 0.24 | 0.003 | 0.006 | 5 | 3 | N(R) | 4 | |
| 2. | cone | $s\cos(t),s\sin(t),s$ | 122 | 73.45 | 0.192 | 0.288 | 98 | 8 | N(R) | 14 | |
| 3. | paraboloid | $s\cos(t),s\sin(t),s^2$ | 122 | 71.60 | 0.192 | 0.296 | 98 | 8 | N(R) | 37 | |
| 4. | surface of revolution | $s\cos(t),s\sin(t),\cos(s)$ | 122 | 71.80 | 0.193 | 0.288 | 98 | 8 | N(R) | 37 | |
| 5. | sphere | $\sin(t)\cos(s),\sin(t)\sin(s),\cos(t)$ | 104148 | halt | 19496.602 | 714.161 | 43018 | 21 | N(R) | 186 | |
| 6. | sphere2 | $\cos(t)\cos(s),\sin(t)\cos(s),\sin(s)$ | 76280 | halt | 4492.977 | 397.157 | 32076 | 95 | N(R) | 776 | |
| 7. | stereographic shpere | $2t/(1 t^2 s^2),2s/(1 t^2 s^2),(t^2 s^2-1)/(1 t^2 s^2)$ | 3540 | 7112.54 | 25.402 | 11.025 | 3126 | 22 | N(R) | 283 | |
| 8. | twisted shpere | $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$ | >1812221 | not computed | not computed | not computed | not computed | not computed | not computed |
Remarks
- ↑ Many thanks to Tatjana Kalinka for providing this list of curves and surfaces.
- ↑ 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.
- ↑ 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.
- ↑ 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.
