Implicitization

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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

supports

289 193.62 0.048 0.452 289 35 N(R) 454
2. cardioid $a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t))$

supports

37 6.52 0.005 0.024 37 10 N(R) 33
3. circle $ \cos(t),\sin(t)$

supports

5 0.004 0.016 0.004 5 3 N(R) 4
4. conchoid $a \cos(t),ah \sin(t)$

supports

12 0.84 0.003 0.008 12 4 N(R) 6
5. ellipse $a\cos(t),b\sin(t)$

supports

5 0.15 0.001 0.004 5 3 N(R) 4
6. folium of descartes $3ah/(1+ h^3), 3ah^2/(ah^3)$

supports

14 0.94 0.004 0.008 14 6 N(R) 10
7. involute of a circle $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$

supports

14 1.00 0.001 0.007 14 6 N(R) 7
8. nephroid $a(3\cos(t)-\cos(3t)),a(3\sin(t)-\sin(3t))$

supports

289 195.27 0.004 0.240 289 35 N(R) 454
9a. plateau curve $a\sin(3t)/\sin(t),2a\sin(2t)$

supports

94 33.02 0.012 0.064 94 15 N(R) 55
9b. plateau curve $a\sin(6t)/ \sin(2t), 2a\sin(4t)$

supports

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 $

supports

1944 3948.80 0.416 2.356 1944 84 N(R) 1600
11. tricuspoid $a(2\cos(t)+\cos(2t)),a(2\sin(t)-\sin(2t))$

supports

37 6.20 0.008 0.024 37 10 N(R) 33
12. witch of agnesi $ah,a/(1+h^2)$

supports

2 0.03 0.007 0.004 2 2 N(R) 2
13. circle (3 systems) $(-t^2 +1)/s, 2t/s, t^2 -s +1$

supports

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 $\cos(t),\sin(t),s$

supports

5 0.24 0.003 0.006 5 3 N(R) 4
2. cone $s\cos(t),s\sin(t),s$

supports

122 73.45 0.192 0.288 98 8 N(R) 14
3. paraboloid $s\cos(t),s\sin(t),s^2$

supports

122 71.60 0.192 0.296 98 8 N(R) 37
4. surface of revolution $s\cos(t),s\sin(t),\cos(s)$

supports

122 71.80 0.193 0.288 98 8 N(R) 37
5. sphere $\sin(t)\cos(s),\sin(t)\sin(s),\cos(t)$

supports

104148 halt 19496.602 714.161 43018 21 N(R) 186
6. sphere2 $\cos(t)\cos(s),\sin(t)\cos(s),\sin(s)$

supports

76280 halt 4492.977 397.157 32076 95 N(R) 776
7. stereographic shpere $2t/(1 t^2 s^2),2s/(1 t^2 s^2),(t^2 s^2-1)/(1 t^2 s^2)$

supports

3540 7112.54 25.402 11.025 3126 22 N(R) 283
8. twisted shpere $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$

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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