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== Implicitization experiments on curves and surfaces == | == Implicitization experiments on curves and surfaces == | ||
Revision as of 23:56, 26 January 2010
[Implicitization experiments on curves and surfaces]
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 | 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)) | 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) | 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) | 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) | 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) | 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)) | 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)) | 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) | 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) | 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 | 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)) | 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) | 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 | 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 | 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 | 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 | 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) | 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) | 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) | 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) | 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)) | >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.
