Implicitization
From ErgaWiki
(→Implicitization) |
(→Implicitization experiments on curves and surfaces) |
||
| Line 8: | Line 8: | ||
Each parametric equation can be transformed to a polynomial. The supports of the polynomials after the Cayley trick are stored in the files in the "supports" column. The file format is as follows: the first line contain the number of points and their dimension and in the following lines each column contains the coordinates of a point. | Each parametric equation can be transformed to a polynomial. The supports of the polynomials after the Cayley trick are stored in the files in the "supports" column. The file format is as follows: the first line contain the number of points and their dimension and in the following lines each column contains the coordinates of a point. | ||
| - | Implicitization using predicted support has been implemented in ''Maple''. [http://ergawiki.di.uoa.gr/experiments/i2D.mpl | + | Implicitization using predicted support has been implemented in ''Maple''. |
| + | In our experiments we use two support prediction methods: one applies only for curves, and another is general. We present our implementation of the [http://ergawiki.di.uoa.gr/experiments/i2D.mpl "curves only"] method and some [http://ergawiki.di.uoa.gr/experiments/i2Dexamples.mw examples]. Here we apply numeric as well as exact methods for linear solving in order to obtain coefficients of the implicit equation. | ||
| + | [http://ergawiki.di.uoa.gr/experiments/impl_gen.mpl Another] ''Maple'' implementation contains two implicitization methods for general support prediction that use exact solving. [http://ergawiki.di.uoa.gr/experiments/i_gen_examples.mw Examples] of running these methods on the curves and surfaces. | ||
| + | |||
| + | For more information see <ref> I. Z. Emiris, T. Kalinka, Ch. Konaxis, Implicitization of curves and surfaces using predicted support, 2011 [http://ergawiki.di.uoa.gr/experiments/Implicitization-snc.pdf] </ref>. | ||
== Support prediction == | == Support prediction == | ||
Revision as of 19:06, 12 May 2011
Contents |
Implicitization experiments on curves and surfaces
These are some experimental results of the algorithms that compute the Newton polytope of the Resultant. For more information see [1].
For the case of curves, there are two equations on one variable t. For the case of surfaces, there are three equations on two variables t,s. Additionally, a,b,c are constants.
Each parametric equation can be transformed to a polynomial. The supports of the polynomials after the Cayley trick are stored in the files in the "supports" column. The file format is as follows: the first line contain the number of points and their dimension and in the following lines each column contains the coordinates of a point.
Implicitization using predicted support has been implemented in Maple. In our experiments we use two support prediction methods: one applies only for curves, and another is general. We present our implementation of the "curves only" method and some examples. Here we apply numeric as well as exact methods for linear solving in order to obtain coefficients of the implicit equation. Another Maple implementation contains two implicitization methods for general support prediction that use exact solving. Examples of running these methods on the curves and surfaces.
For more information see [2].
Support prediction
Curves
| No | curve [3] | parametric equation | supports | # mixed subdivisions |
Enum by reverse search (sec) [4] |
TOPCOM point2alltriang (sec) [5] |
TOPCOM point2triang(sec) [6] | # mixed cell configurations | # N(R) vertices | N(R) vertices | # all lattice points of N(R) | implicit polytope's terms & coefficients | implicit support prediction (for curves) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1. | astroid | a\cos(t)^3,a\sin(t)^3 | 289 | 193.62 | 0.048 | 0.452 | 289 | 35 | N(R) | 454 | implicit | support | |
| 2. | cardioid | $a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t))$ | 37 | 6.52 | 0.005 | 0.024 | 37 | 10 | N(R) | 33 | implicit | support | |
| 3. | circle | $ \cos(t),\sin(t)$ | 5 | 0.004 | 0.016 | 0.004 | 5 | 3 | N(R) | 4 | implicit | support | |
| 4. | conchoid | $a+\cos(t),a\tan(t)+\sin(t)$ | 12 | 0.84 | 0.003 | 0.008 | 12 | 4 | N(R) | 6 | implicit | support | |
| 5. | ellipse | $a\cos(t),b\sin(t)$ | 5 | 0.15 | 0.001 | 0.004 | 5 | 3 | N(R) | 4 | implicit | support | |
| 6. | folium of Descartes | $3at/(1+t^3),3at^2/(1+t^3)$ | 14 | 0.94 | 0.004 | 0.008 | 14 | 6 | N(R) | 10 | implicit | support | |
| 7. | involute of a circle | $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$ | 14 | 1.00 | 0.001 | 0.007 | 14 | 6 | N(R) | 7 | implicit | ||
| 8. | nephroid | $a(3\cos(t)-\cos(3t)),a(3\sin(t)-\sin(3t))$ | 289 | 195.27 | 0.004 | 0.240 | 289 | 35 | N(R) | 454 | implicit | support | |
| 9a. | Plateau curve | $a\sin(3t)/\sin(t),2a\sin(2t)$ | 94 | 33.02 | 0.012 | 0.064 | 94 | 15 | N(R) | 55 | implicit | ||
| 9b. | plateau curve | $a\sin(6t)/ \sin(2t), 2a\sin(4t)$ | 42168 | halt | 25.934 | 85.597 | 42168 | 495 | N(R) | not computed | implicit | ||
| 10. | talbot's curve | $(a^2 + c^2 \sin( t)^2) \cos( t)/a, (a^2 - 2c^2 + c^2\sin(t)^2)\sin(t)/b $ | 1944 | 3948.80 | 0.416 | 2.356 | 1944 | 84 | N(R) | 1600 | implicit | ||
| 11. | tricuspoid | $a(2\cos(t)+\cos(2t)),a(2\sin(t)-\sin(2t))$ | 37 | 6.20 | 0.008 | 0.024 | 37 | 10 | N(R) | 33 | implicit | support | |
| 12. | witch of Agnesi | $at,a/(1 + t^2)$ | 2 | 0.03 | 0.007 | 0.004 | 2 | 2 | N(R) | 2 | implicit | ||
| 13. | circle (3 systems) | $(-t^2 +1)/s, 2t/s, t^2 -s +1$ | 26 | 6.00 | 0.020 | 0.052 | 26 | 6 | N(R) | 7 | implicit | ||
| 14. | A' Campo curve | $9ts-4,t^3+s^2+c1,t^2+s^3+c2$ | 29 | - | - | - | 29 | 6 | N(R) | 18 | implicit |
Surfaces
| No | surface | parametric equation | supports | # mixed subdivisions |
Enum by reverse search (sec) |
TOPCOM point2alltriang (sec) |
TOPCOM point2triang(sec) | # mixed cell configurations | # N(R) vertices | N(R) vertices | # all lattice points of N(R) | implicit polytope's terms & coefficients |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1. | cylinder | $\cos(t),\sin(t),s$ | 5 | 0.24 | 0.003 | 0.006 | 5 | 3 | N(R) | 4 | implicit | |
| 2. | cone | $s\cos(t),s\sin(t),s$ | 122 | 73.45 | 0.192 | 0.288 | 98 | 8 | N(R) | 14 | implicit | |
| 3. | paraboloid | $s\cos(t),s\sin(t),s^2$ | 122 | 71.60 | 0.192 | 0.296 | 98 | 8 | N(R) | 37 | implicit | |
| 4. | surface of revolution | $s\cos(t),s\sin(t),\cos(s)$ | 122 | 71.80 | 0.193 | 0.288 | 98 | 8 | N(R) | 37 | implicit | |
| 5. | sphere | $\sin(t)\cos(s),\sin(t)\sin(s),\cos(t)$ | 104148 | halt | 19496.602 | 714.161 | 43018 | 21 | N(R) | 186 | implicit | |
| 6. | sphere2 | $\cos(t)\cos(s),\sin(t)\cos(s),\sin(s)$ | 76280 | halt | 4492.977 | 397.157 | 32076 | 95 | N(R) | 776 | implicit | |
| 7. | stereographic sphere | $2t/(1 t^2 s^2),2s/(1 t^2 s^2),(t^2 s^2-1)/(1 t^2 s^2)$ | 3540 | 7112.54 | 25.402 | 11.025 | 3126 | 22 | N(R) | 283 | implicit | |
| 8. | twisted sphere | $a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$ | >1812221 | not computed | not computed | not computed | not computed | not computed | not computed | implicit |
Implicitization
These results have been obtained using Maple 11 exact solving methods.
| No | surface | parametric degree | parametric terms | "curves only" matrix size |
"curves only" method runtime (sec) | generic method: matrix size |
generic method runtime (sec) | generic method nonzero coeffs | "mapping" method: matrix size |
"mapping" method runtime (sec) | "mapping" method nonzero coeffs (actual coeffs) | implicit degree |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1. | cardioid | 4, 4 | 3, 4 | 15 | 0.128 | 33 | 0.248 | 33 | 25 | 0.656 | 7 | 4 |
| 2. | conhoid | 2, 3 | 2, 4 | 15 | 0.094 | 6 | 0.096 | 6 | 15 | 0.308 | 6 | 4 |
| 3. | folium of descartes | 3, 3 | 3, 3 | 10 | 0.092 | 5 | 0.032 | 3 | 11 | 0.144 | 3 | 3 |
| 4. | nephroid | 4, 4 | 4, 5 | 28 | 0.256 | 426 | not computed | not computed | 49 | 5.452 | 10 | 6 |
| 5. | ranunculoid | 12, 12 | 7, 12 | 91 | 8809.43 | not computed | not computed | not computed | not computed | not computed | not computed | not computed |
| 6. | talbot's curve | 6, 6 | 3, 4 | 28 | 109.342 | 421 | not computed | not computed | not computed | not computed | 23 | 6 |
| 7. | tricuspoid | 4, 4 | 3, 4 | 15 | 0.216 | 33 | 0.236 | 33 | 25 | 0.540 | 8 | 4 |
| 8. | whitch of agnesi | 1, 2 | 2, 2 | 4 | 0.016 | 2 | 0.044 | 2 | 4 | 0.048 | 3 | 3 |
Remarks
[1] Fisikopoulos Vissarion, Triangulations of point sets, high dimensional Polytopes and Applications, Master thesis, University of Athens, Graduate Program in Logic, Algorithms and Computation, 2010 [1]
[2] I. Z. Emiris, T. Kalinka, Ch. Konaxis, Implicitization of curves and surfaces using predicted support, 2011 [2]
[3] Many thanks to Tatjana Kalinka for providing this list of curves and surfaces.
[4] 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.
[5] 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.
[6] 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.
