Implicitization
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The former describes the geometric object as the set of values of parametric expressions over a set of values of the parameters. | 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. | 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. | ||
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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. | 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. | ||
At this point, our first approach was to compute the Newton polytope of the sparse resultant, or resultant polytope, and then to recover from this polytope the Newton polytope of the implicit equation, or implicit polytope. This method was very inefficient as it can be seen in the examples below. We follow a more direct approach and compute a projection of the resultant's Newton polytope, without computing the resultant polytope itself. This projection gives information on the implicit equation's support and it 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. | At this point, our first approach was to compute the Newton polytope of the sparse resultant, or resultant polytope, and then to recover from this polytope the Newton polytope of the implicit equation, or implicit polytope. This method was very inefficient as it can be seen in the examples below. We follow a more direct approach and compute a projection of the resultant's Newton polytope, without computing the resultant polytope itself. This projection gives information on the implicit equation's support and it 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. | ||
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The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [[EFKP]http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108]. | The first section contains experimental results on the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [[EFKP]http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108]. | ||
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== Implicitization using predicted support has been implemented in ''Maple''. | == 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 [http://ergawiki.di.uoa.gr/experiments/simpl.mpl '''''Maple'' implementation'''] of the implicitization with predicted support and some [http://ergawiki.di.uoa.gr/experiments/examples.mw examples]. Here we apply numeric (SVD) as well as exact methods for linear solving | + | 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 [http://ergawiki.di.uoa.gr/experiments/simpl.mpl '''''Maple'' implementation'''] of the implicitization with predicted support and some [http://ergawiki.di.uoa.gr/experiments/examples.mw examples]. Here we apply numeric (SVD) as well as exact methods for linear solving in order to obtain coefficients of the implicit equation. |
| - | For | + | For background information see [[EKKL-TCS]http://ergawiki.di.uoa.gr/experiments/EBKKtcs11.pdf]. For recent progress see [[EKK2015]http://www.sciencedirect.com/science/article/pii/S1524070315000260]. |
== Support prediction == | == Support prediction == | ||
Revision as of 22:05, 14 December 2015
Contents |
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. At this point, our first approach was to compute the Newton polytope of the sparse resultant, or resultant polytope, and then to recover from this polytope the Newton polytope of the implicit equation, or implicit polytope. This method was very inefficient as it can be seen in the examples below. We follow a more direct approach and compute a projection of the resultant's Newton polytope, without computing the resultant polytope itself. This projection gives information on the implicit equation's support and it 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 the support prediction step which is the computation of the resultant polytope, or its projection, given the system of polynomials constructed by the parametric expressions. We developed an output-sensitive algorithm for computing resultant polytopes and their projections [[EFKP]http://www.worldscientific.com/doi/abs/10.1142/S0218195913600108]. Its implementation, called ResPol can be found in [[Sourceforge]https://sourceforge.net/projects/respol/]. Let us demonstrate how to prepare the input for ResPol, given the parametric equations of the surface:
x = 1/2*s^2 - 1/2*t^2 - 1/4*s^4 + 3/2*s^2*t^2 - 1/4*t^4,
y = -s*t - s^3*t + s*t^3,
z = 2/3*s^3 - 2*s*t^2.
From the parametric expressions we define the following three polynomials:
F_1 = x- 1/2*s^2 + 1/2*t^2 + 1/4*s^4 - 3/2*s^2*t^2 + 1/4*t^4,
F_2 = y + s*t + s^3*t - s*t^3,
F_3 = z - 2/3*s^3 + 2*s*t^2.
Their supports with respect to the variables s,t are:
A_1 = [[0,0],[2,0],[0,2],[4,0],[2,2],[0,4]],
A_2 = [[0,0],[1,1],[3,1],[1,3]],
A_3 = [[0,0],[3,0],[1,2]].
The input to ResPol is as follows:
2
6 4 3 | 0 6 10
[[0,0],[2,0],[0,2],[4,0],[2,2],[0,4],[0,0],[1,1],[3,1],[1,3],[0,0],[3,0],[1,2]],
where the first line is the dimension of the supports (equal to the number of parameters), the second line contains the cardinalities of the supports and the elements of each support that are exponents of monomials whose coefficients contain an implicit variable (x,y, or z). In this example, these monomials correspond to the [0,0] point from each support. The latter define the projection of the resultant polytope thet ResPol computes. In the case of implicitization of polynomial parametric curves or surfaces, we can safely omit defining these points in the input file. Note that we number the elements in each support continuously starting from 0.
Then, Respol predicts the following vertices of the implicit polytope:
[3,2,2],[9,0,4],[0,12,0],[0,0,16],[4,4,0],[0,0,6],[8,4,0],[0,8,0],[3,0,4],[0,2,4],[3,2,2].
== 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 background information see [[EKKL-TCS]http://ergawiki.di.uoa.gr/experiments/EBKKtcs11.pdf]. For recent progress see [[EKK2015]http://www.sciencedirect.com/science/article/pii/S1524070315000260].
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 | 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 |
References
1 I.Z.Emiris, V.Fisikopoulos, C.Konaxis, L.Peñaranda. An oracle-based, output sensitive algorithm for projections of resultant polytopes. International Journal of Computational Geometry and Applications vol. 23, pp. 397-423, (Special issue) [World Scientific[1]] 2013.
2 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/
3 I. Z. Emiris, T. Kalinka, Ch. Konaxis, T. Luu Ba, Implicitization of curves and surfaces using predicted support, 2011. Theor. Comp. Sci..
6 This is the computation time of enumeration of regular triangulations algorithm using reverse search. We would like to thank Fumihiko TAKEUCHI for running the experiments and providing the 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.
