Implicitization
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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>. | 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>. | ||
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== Support prediction == | == Support prediction == | ||
Revision as of 12:13, 2 March 2012
Contents |
Implicitization experiments on curves and surfaces
Implicitization is the change of parametric representation to an implicit, or algebraic, representation. The former describes the geometric object as the set of values of parametric expressions over a set of values of the parameters. The latter describes the same object as the zero-set of a single polynomial. Example: the circle below is given parametrically by x=cos(t), y=sin(t); its implicit equation is x^2+y^2=1.
For the case of curves, there are two equations in one parameter t. For the case of surfaces, there are 3 expressions in two parameters t,s. Additionally, in certain parameterizations, we consider families of curves/surfaces with respect to free constants a,b,c.
Each parametric expression can be transformed to a polynomial. We consider the system of polynomials in the parameters and define its sparse resultant. We compute the resultant's Newton polytope, which gives information on the implicit equation's support. Then, we compute the implicit equation by interpolation, which amounts to linear algebra.
These are some experimental results of the algorithms that compute the Newton polytope of the Sparse Resultant. For more information see [1].
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].
References
- ↑ I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. An output-sensitive algorithm for computing projections of resultant polytopes. To appear in 28th Annual Symposium on Computational Geometry (SoCG 2012), Chapel Hill, NC, USA. (available from arXiv:1108.5985) [1]
- ↑ I. Z. Emiris, T. Kalinka, Ch. Konaxis, Implicitization of curves and surfaces using predicted support, 2011 [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.
Curves
| No | curve | 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 |
Surfaces
| No | surface | parametric degree | parametric terms | # inside points | 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. | Infinite cylinder | 2, 2, 1 | 2, 3, 2 | 4 | 4 | 0.036 | 4 | 9 | 0.072 | 3 | 2 |
| 2. | Hyperbolic paraboloid | 1, 1, 2 | 2, 2, 3 | 3 | 3 | 0.028 | 3 | 7 | 0.064 | 3 | 2 |
| 3. | Infinite cone | 3, 2, 1 | 4, 3, 2 | 14 | 8 | 0.040 | 6 | 19 | 0.224 | 3 | 4 |
| 4. | Whitney umbrella | 2, 1, 2 | 2, 2, 2 | 2 | 2 | 0.024 | 2 | 4 | 0.056 | 2 | 3 |
| 5. | Monkey saddle | 1, 1, 3 | 2, 2, 3 | 3 | 3 | 0.028 | 3 | 8 | 0.064 | 3 | 3 |
| 6. | Handkerchief surface | 1, 1, 3 | 2, 2, 5 | 2 | 2 | 0.020 | 2 | 4 | 0.036 | 5 | 3 |
| 7. | Crossed surface | 1, 1, 4 | 2, 2, 2 | 5 | 5 | 0.032 | 5 | 10 | 0.072 | 2 | 4 |
| 8. | Quartoid | 1, 1, 4 | 2, 2, 4 | 4 | 4 | 0.012 | 4 | 16 | 0.140 | 4 | 4 |
| 9. | Peano surface | 1, 1, 4 | 2, 2, 4 | 4 | 4 | 0.020 | 4 | 10 | 0.080 | 4 | 4 |
| 10. | Bohemian dome | 2, 4, 2 | 2, 6, 3 | 142 | 58 | 2.764 | error | 125 | 83.362 | 7 | 4 |
| 11. | Swallowtail surface | 4, 3, 1 | 3, 3, 2 | 12 | 12 | 0.052 | 12 | 25 | 0.432 | 6 | 5 |
| 12. | Sine surface | 2, 2, 4 | 3, 3, 8 | 1027 | 87 | 9.244 | 66 | 125 | 107.102 | 7 | 6 |
| 13. | Enneper's surface | 3, 3, 2 | 4, 3, 3 | 439 | 258 | 74.304 | 258 | 106 | 33.562 | 57 | 9 |
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 [3]
[2] I. Z. Emiris, T. Kalinka, Ch. Konaxis, Implicitization of curves and surfaces using predicted support, 2011 [4]
[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.
