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| - | [Implicitization experiments on curves and surfaces] | + | [[File:implicitization.jpg|frameless|link=Implicitization]] |
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| | [[Implicitization experiments on curves and surfaces ]] | | [[Implicitization experiments on curves and surfaces ]] |
| - |
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| - | __TOC__
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| - |
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| - | == Implicitization experiments ==
| |
| - | === Curves ===
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| - | {| class="experiments sortable"
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| - | ! No
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| - | ! class="unsortable" | curve <ref> Many thanks to Tatjana Kalinka for providing this list of curves and surfaces. </ref>
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| - | ! class="unsortable" | equation
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| - | ! class="unsortable" | supports
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| - | ! <nowiki># mixed subdivisions</nowiki>
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| - | ! class="run" |
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| - | Enum by reverse search (sec) <ref> This is the computation time of [http://jn.wspc.com.sg/google/pdf/S0218195902000980.pdf enumeration of regular triangulations algorithm using reverse search]. I would like to thank very much [http://www.purple.dti.ne.jp/pub/cv.html Fumihiko TAKEUCHI] for running the experiments and providing this results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.</ref>
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| - | ! class="run" |
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| - | TOPCOM point2alltriang (sec) <ref> This is the computation time of '''points2alltriangs''' 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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| - | ! class="run" |
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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>
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| - | ! class="unsortable" | <nowiki># mixed cell configurations</nowiki>
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| - | ! class="unsortable" | <nowiki># extreme terms</nowiki>
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| - | ! class="unsortable" | <nowiki># all terms</nowiki>
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| - | |-
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| - | |
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| - | |-
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| - | | 1.
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| - | || astroid
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| - | ||<math>a\cos(t)^3,a\sin(t)^3</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports1.dat supports]
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| - | | class="topcom" | 289
| |
| - | | 193.62
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| - | | 0.048
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| - | | 0.452
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| - | | 289
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| - | | 35
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| - | | 454
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| - | |-
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| - | | 2.
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| - | | cardioid
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| - | | <math>a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t))</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports2.dat supports]
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| - | | class="topcom" | 37
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| - | | 6.52
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| - | | 0.005
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| - | | 0.024
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| - | | 37
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| - | | 10
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| - | | 33
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| - | |-
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| - | | 3.
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| - | | circle
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| - | |<math> \cos(t),\sin(t)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports3.dat supports]
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| - | | class="topcom" | 5
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| - | | 0.004
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| - | | 0.016
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| - | | 0.004
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| - | | 5
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| - | | 3
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| - | | 4
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| - | |-
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| - | | 4.
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| - | | conchoid
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| - | | <math>a \cos(t),ah \sin(t)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports4.dat supports]
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| - | | class="topcom" | 12
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| - | | 0.84
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| - | | 0.003
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| - | | 0.008
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| - | | 12
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| - | | 4
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| - | | 6
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| - | |-
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| - | | 5.
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| - | | ellipse
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| - | | <math>a\cos(t),b\sin(t)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports5.dat supports]
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| - | | class="topcom" | 5
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| - | | 0.15
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| - | | 0.001
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| - | | 0.004
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| - | | 5
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| - | | 3
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| - | | 4
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| - | |-
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| - | | 6.
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| - | | folium of descartes
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| - | | <math>3ah/(1+ h^3), 3ah^2/(ah^3)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports6.dat supports]
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| - | | class="topcom" | 14
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| - | | 0.94
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| - | | 0.004
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| - | | 0.008
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| - | | 14
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| - | | 6
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| - | | 10
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| - | |-
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| - | | 7.
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| - | | involute of a circle
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| - | | <math>a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports7.dat supports]
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| - | | class="topcom" | 14
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| - | | 1.00
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| - | | 0.001
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| - | | 0.007
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| - | | 14
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| - | | 6
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| - | | 7
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| - | |-
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| - | | 8.
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| - | | nephroid
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| - | | <math>a(3\cos(t)-\cos(3t)),a(3\sin(t)-\sin(3t))</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports8.dat supports]
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| - | | class="topcom" | 289
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| - | | 195.27
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| - | | 0.004
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| - | | 0.240
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| - | | 289
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| - | | 35
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| - | | 454
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| - | |-
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| - | | 9a.
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| - | | plateau curve
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| - | | <math>a\sin(3t)/\sin(t),2a\sin(2t)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports9a.dat supports]
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| - | | class="topcom" | 94
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| - | | 33.02
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| - | | 0.012
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| - | | 0.064
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| - | | 94
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| - | | 15
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| - | | 55
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| - | |-
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| - | | 9b.
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| - | | plateau curve
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| - | | <math>a\sin(6t)/ \sin(2t), 2a\sin(4t)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports9b.dat supports]
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| - | | class="topcom" | 42168
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| - | | class="halt" | halt
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| - | | 25.934
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| - | | 85.597
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| - | | 42168
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| - | | 495
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| - | | class="halt" | not computed
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| - | |-
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| - | | 10.
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| - | | talbot's curve
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| - | | <math>(a^2 + f^2 \sin( t)^2) \cos( t)/a, (a^2 - 2f^2 + (f^2)\sin(t)^2)\sin(t)/b </math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports10.dat supports]
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| - | | class="topcom" | 1944
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| - | | 3948.80
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| - | | 0.416
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| - | | 2.356
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| - | | 1944
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| - | | 84
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| - | | 1600
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| - | |-
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| - | | 11.
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| - | | tricuspoid
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| - | | <math>a(2\cos(t)+\cos(2t)),a(2\sin(t)-\sin(2t))</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports11.dat supports]
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| - | | class="topcom" | 37
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| - | | 6.20
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| - | | 0.008
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| - | | 0.024
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| - | | 37
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| - | | 10
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| - | | 33
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| - | |-
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| - | | 12.
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| - | | witch of agnesi
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| - | | <math>ah,a/(1 h^2)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports12.dat supports]
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| - | | class="topcom" | 2
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| - | | 0.03
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| - | | 0.007
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| - | | 0.004
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| - | | 2
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| - | | 2
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| - | | 2
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| - | |-
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| - | | 13.
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| - | | circle (3 systems)
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| - | | <math>(-t^2 +1)/s, 2t/s, t^2 -s +1</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/curves_supports/supports13.dat supports]
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| - | | class="topcom" | 26
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| - | | 6.00
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| - | | 0.020
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| - | | 0.052
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| - | | 26
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| - | | 6
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| - | | 7
| |
| - | |}
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| - |
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| - | === Surfaces ===
| |
| - | {|class="experiments sortable"
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| - | ! No
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| - | ! surface
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| - | ! class="unsortable" | equation
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| - | ! class="unsortable" | supports
| |
| - | ! <nowiki># mixed subdivisions</nowiki>
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| - | ! class="run" |
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| - | Enum by reverse search (sec)
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| - | ! class="run" |
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| - | TOPCOM point2alltriang (sec)
| |
| - | ! class="run" |
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| - | TOPCOM point2triang(sec)
| |
| - | ! <nowiki># mixed cell configurations</nowiki>
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| - | ! <nowiki># extreme terms</nowiki>
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| - | ! <nowiki># all terms</nowiki>
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| - | |-
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| - | |
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| - | |-
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| - | | 1.
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| - | | cylinder
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| - | | <math>\cos(t),\sin(t),s</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports1.dat supports]
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| - | | class="topcom" | 5
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| - | | 0.24
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| - | | 0.003
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| - | | 0.006
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| - | | 5
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| - | | 3
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| - | | 4
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| - | |-
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| - | | 2.
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| - | | cone
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| - | | <math>s\cos(t),s\sin(t),s</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports2.dat supports]
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| - | | class="topcom" | 122
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| - | | 73.45
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| - | | 0.192
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| - | | 0.288
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| - | | 98
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| - | | 8
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| - | | 14
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| - | |-
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| - | | 3.
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| - | | paraboloid
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| - | | <math>s\cos(t),s\sin(t),s^2</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports3.dat supports]
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| - | | class="topcom" | 122
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| - | | 71.60
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| - | | 0.192
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| - | | 0.296
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| - | | 98
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| - | | 8
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| - | | 37
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| - | |-
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| - | | 4.
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| - | | surface of revolution
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| - | | <math>s\cos(t),s\sin(t),\cos(s)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports4.dat supports]
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| - | | class="topcom" | 122
| |
| - | | 71.80
| |
| - | | 0.193
| |
| - | | 0.288
| |
| - | | 98
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| - | | 8
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| - | | 37
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| - | |-
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| - | | 5.
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| - | | sphere
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| - | | <math>\sin(t)\cos(s),\sin(t)\sin(s),\cos(t)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports5.dat supports]
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| - | | class="topcom" | 104148
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| - | | class="halt" | halt
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| - | | 19496.602
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| - | | 714.161
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| - | | 43018
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| - | | 21
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| - | | 186
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| - | |-
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| - | | 6.
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| - | | sphere2
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| - | | <math>\cos(t)\cos(s),\sin(t)\cos(s),\sin(s)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports6.dat supports]
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| - | | class="topcom" | 76280
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| - | | class="halt" | halt
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| - | | 4492.977
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| - | | 397.157
| |
| - | | 32076
| |
| - | | 95
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| - | | 776
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| - | |-
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| - | | 7.
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| - | | stereographic shpere
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| - | | <math>2t/(1 t^2 s^2),2s/(1 t^2 s^2),(t^2 s^2-1)/(1 t^2 s^2)</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports7.dat supports]
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| - | | class="topcom" | 3540
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| - | | 7112.54
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| - | | 25.402
| |
| - | | 11.025
| |
| - | | 3126
| |
| - | | 22
| |
| - | | 283
| |
| - | |-
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| - | | 8.
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| - | | twisted shpere
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| - | | <math>a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))</math>
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| - | |
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| - | [http://ergawiki.di.uoa.gr/surfaces_supports/supports8.dat supports]
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| - | | class="topcom" | >1812221
| |
| - | | class="halt" | not computed
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| - | | class="halt" | not computed
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| - | | class="halt" | not computed
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| - | | class="halt" | not computed
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| - | | class="halt" | not computed
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| - | | class="halt" | not computed
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| - | |}
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| - |
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| - | == Remarks ==
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| - |
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| - | <references />
| |
