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== Implicitization experiments on curves and surfaces ==
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Welcome to the Wiki Pages of ΕρΓΑ Lab (Algebraic and Geometric Algorithms) at University of Athens [http://erga.di.uoa.gr (Lab webpage)]
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{| class="experiments sortable"
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[[File:implicitization.jpg|frameless|link=Implicitization]]
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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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[curves_supports/supports1.dat supports]
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| class="topcom" | 289
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| 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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[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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[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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[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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[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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[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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[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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[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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[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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[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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[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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[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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[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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[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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 +
[[Implicitization | Implicitization experiments on curves and surfaces.]]
 +
Implicitization is the problem of transforming the parametric representation of a geometric object (e.g. curve of surface) to its implicit representation. Our group is working on matrix representations of implicit varieties, by exploiting the structure of the polynomials.
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{|class="experiments sortable"
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(figure of the Newton polytope of a sparse resultant, used for sparse implicitization)
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! No
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! surface
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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)
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! class="run" |
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TOPCOM point2alltriang (sec)
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! class="run" |
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TOPCOM point2triang(sec)
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! <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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[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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[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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[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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[surfaces_supports/supports4.dat supports]
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| class="topcom" | 122
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| 71.80
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| 0.193
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| 0.288
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| 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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[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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[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
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| 32076
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| 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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[surfaces_supports/supports7.dat supports]
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| class="topcom" | 3540
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| 7112.54
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| 25.402
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| 11.025
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| 3126
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| 22
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| 283
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|-
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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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[surfaces_supports/supports8.dat supports]
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| class="topcom" | &gt;1812221
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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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| class="halt" | not computed
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|}
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== Remarks ==
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<references />
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Current revision as of 21:29, 16 December 2015

Welcome to the Wiki Pages of ΕρΓΑ Lab (Algebraic and Geometric Algorithms) at University of Athens (Lab webpage)

Implicitization experiments on curves and surfaces. Implicitization is the problem of transforming the parametric representation of a geometric object (e.g. curve of surface) to its implicit representation. Our group is working on matrix representations of implicit varieties, by exploiting the structure of the polynomials.

(figure of the Newton polytope of a sparse resultant, used for sparse implicitization)

Retrieved from "http://ergawiki.di.uoa.gr/index.php/Main_Page"
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