Implicitization

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Implicitization experiments on curves and surfaces

Implicitization is the change of representation of a (hyper)surface, from a parametric one 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 unit circle is given parametrically by x=cos(t), y=sin(t); its implicit equation is x^2+y^2=1. We shall call x,y the implicit variables.


For the case of curves, there are two parametric equations in one parameter. For the case of surfaces, there are 3 parametric expressions in two parameters. Each parametric expression can be transformed to a polynomial. For example, the expression x=f(t)/g(t), is transformed to the polynomial F(t)=x*g(t)-f(t). We consider the system of all such polynomials and define its sparse resultant. We compute a projection of the resultant's Newton polytope, which gives information on the implicit equation's support. The projection is defined by the coefficients of the polynomials which involve implicit variables. Then, we compute the implicit equation by interpolation, which amounts to linear algebra.


The first section contains experimental results on support prediction step which is the computation of the Newton polytope of the sparse resultant, the resultant polytope, given the system of polynomials constructed by the parametric expressions. 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. The method used here retrieve the resultant polytope from the computation of another polytope called secondary polytope. We use TOPCOM [1] to compute the vertices of the secondary polytope each of which gives us a vertex of the resultant polytope. The secondary polytope has much more vertices that the resultant polytope as shown in the experiments below. To overcome the inefficiency of computing the secondary polytope an output-sensitive algorithm for computing resultant polytopes and its projections has been proposed in [2]. Its implementation can be found in [3].

Implicitization using predicted support has been implemented in Maple. In our experiments we use two support prediction methods: one applies only for curves and can be fully implemented in Maple, another is general (implemented in respol). We present our Maple implementation of the implicitization with predicted support and some examples. Here we apply numeric (SVD) as well as exact methods for linear solving in order to obtain coefficients of the implicit equation.

For more information see [4], [5].

Support prediction

Curves

No curve parametric equation supports # mixed subdivisions

Enum by reverse search (sec) [6]

TOPCOM point2alltriang (sec) [7]

TOPCOM point2triang(sec) [8]

# 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

supports

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

supports

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

supports

5 0.004 0.016 0.004 5 3 N(R) 4 implicit support
4. conchoid $a+\cos(t),a\tan(t)+\sin(t)$

supports

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

supports

5 0.15 0.001 0.004 5 3 N(R) 4 implicit support
6. folium of Descartes $3at/(1+t^3),3at^2/(1+t^3)$

supports

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

supports

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

supports

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

supports

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

supports

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 $

supports

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

supports

37 6.20 0.008 0.024 37 10 N(R) 33 implicit support
12. witch of Agnesi $at,a/(1 + t^2)$

supports

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

supports

26 6.00 0.020 0.052 26 6 N(R) 7 implicit
14. A' Campo curve $9ts-4,t^3+s^2+c1,t^2+s^3+c2$

supports

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

supports

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

supports

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

supports

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

supports

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

supports

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

supports

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

supports

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

supports

>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

References

1 J. Rambau. TOPCOM: Triangulations of point configurations and oriented matroids. In A.M. Cohen, X.-S. Gao, and N. Takayama, editors, Math. Software: ICMS, pages 330–340. World Scientific, 2002.

2 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) http://arxiv.org/abs/1108.5985

3 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. ResPol: A software to compute a projection of the Newton polytope of the resultant, https://sourceforge.net/projects/respol/

4 I. Z. Emiris, T. Kalinka, Ch. Konaxis, Implicitization of curves and surfaces using predicted support, 2011.

5 I. Z. Emiris, T. Kalinka, Ch. Konaxis, Implicitization of curves and surfaces using predicted support, 2011. Submitted to journal.

6 This is the computation time of enumeration of regular triangulations algorithm using reverse search. We 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.

7 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.

8 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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