Implicitization
From ErgaWiki
(Difference between revisions)
(→Curves) |
(→Surfaces) |
||
| Line 233: | Line 233: | ||
TOPCOM point2triang(sec) | TOPCOM point2triang(sec) | ||
! <nowiki># mixed cell configurations</nowiki> | ! <nowiki># mixed cell configurations</nowiki> | ||
| - | ! <nowiki># extreme terms</nowiki> | + | ! <nowiki># N(R) extreme terms</nowiki> |
| + | ! <nowiki>N(R) extreme terms</nowiki> | ||
! <nowiki># all terms</nowiki> | ! <nowiki># all terms</nowiki> | ||
|- | |- | ||
| Line 249: | Line 250: | ||
| 5 | | 5 | ||
| 3 | | 3 | ||
| + | | [http://ergawiki.di.uoa.gr/surfaces_res_extreme/res1.dat N(R)] | ||
| 4 | | 4 | ||
|- | |- | ||
| Line 262: | Line 264: | ||
| 98 | | 98 | ||
| 8 | | 8 | ||
| + | | [http://ergawiki.di.uoa.gr/surfaces_res_extreme/res2.dat N(R)] | ||
| 14 | | 14 | ||
|- | |- | ||
| Line 275: | Line 278: | ||
| 98 | | 98 | ||
| 8 | | 8 | ||
| + | | [http://ergawiki.di.uoa.gr/surfaces_res_extreme/res3.dat N(R)] | ||
| 37 | | 37 | ||
|- | |- | ||
| Line 288: | Line 292: | ||
| 98 | | 98 | ||
| 8 | | 8 | ||
| + | | [http://ergawiki.di.uoa.gr/surfaces_res_extreme/res4.dat N(R)] | ||
| 37 | | 37 | ||
|- | |- | ||
| Line 301: | Line 306: | ||
| 43018 | | 43018 | ||
| 21 | | 21 | ||
| + | | [http://ergawiki.di.uoa.gr/surfaces_res_extreme/res5.dat N(R)] | ||
| 186 | | 186 | ||
|- | |- | ||
| Line 314: | Line 320: | ||
| 32076 | | 32076 | ||
| 95 | | 95 | ||
| + | | [http://ergawiki.di.uoa.gr/surfaces_res_extreme/res6.dat N(R)] | ||
| 776 | | 776 | ||
|- | |- | ||
| Line 327: | Line 334: | ||
| 3126 | | 3126 | ||
| 22 | | 22 | ||
| + | | [http://ergawiki.di.uoa.gr/surfaces_res_extreme/res7.dat N(R)] | ||
| 283 | | 283 | ||
|- | |- | ||
| Line 340: | Line 348: | ||
| class="halt" | not computed | | class="halt" | not computed | ||
| class="halt" | not computed | | class="halt" | not computed | ||
| + | | | ||
| class="halt" | not computed | | class="halt" | not computed | ||
|} | |} | ||
Revision as of 16:53, 1 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 | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | 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 | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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 | N(R) | 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.
