[Math] How to convert the parametric equation into implicit form

groebner-basisparametricplane-curvesrational-functionssymbolic computation

This problem is generated from another Green's theorem related question of mine.

The original equation of the plane curve is not in rational parametric form.

In order to calculate the symbolic solutions of the intersections of the curve itself, I thought its implicit equation form might be helpful or whatever. Nevertheless, such a conversion is interesting itself.

Two articles online are helpful:

  1. http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.204.9611
  2. http://www.mmrc.iss.ac.cn/~xgao/paper/gao-par.pdf

So after simple treatment, I have the following rational parametric equation of the same plane curve as in another problem:

$$\left\{\;
\begin{array}{rl}
x=& \dfrac{2 t \left(3 t^4+50 t^2-33\right)}{\left(t^2+1\right)^3} \\
y=& \dfrac{2 \left(7 t^6-60 t^4+15 t^2+2\right)}{\left(t^2+1\right)^3} \\
\end{array}
\right.$$

How can I convert it into implicit form :$F(x,y)=0$? My difficulty mainly lies in how to implement it via different computer algebra systems, i.e., Mathematica, Maple or whatever.

The answer by Macaulay2 is definitely acceptable, however, there are less documents for beginners on the commands than Maple or Mathematica.

Best Answer

In Macaulay2

R=QQ[s,t,x,y,z,MonomialOrder=>Eliminate 2]
I=ideal(x-2*s*t*(3*t^4+50*t^2*s^2-33*s^4),y-2*(7*t^6-60*t^4*s^2+15*t^2*s^4+2*s^6),z-(t^2+s^2)^3)
gens gb I

yields a term free of $s, t$:

$625x^6+1875x^4y^2+1875x^2y^4+625y^6-182250x^4yz+364500x^2y^3z-36450y^5z+585816x^4z^2+1171632x^2y^2z^2+585816y^4z^2-41620992x^2z^4-41620992y^2z^4+550731776z^6$

Dehomogenising $z$ we get:

$625x^6+1875x^4y^2+1875x^2y^4+625y^6-182250x^4y+364500x^2y^3-36450y^5+585816x^4+1171632x^2y^2+585816y^4-41620992x^2-41620992y^2+550731776=0$

Which plots to the same as your parameterizationplot

Edit

I didn't have to use the homogenised versions of everything, because the more straightforward clearing of denominators (see exercises 3.3.13-14 of Ideals, Varieties and Algorithms)

R=QQ[t,x,y]
I=ideal((t^2+1)^3*x-2*t*(3*t^4+50*t^2-33),(t^2+1)^3*y-2*(7*t^6-60*t^4+15*t^2+2))
gens gb I

gives the relation directly.

Interesting (to me at least) is that the original $x=−9\sin(2t)−5\sin(3t), y=9\cos(2t)−5\cos(3t)$ also satisfies this relation.

Edit2

Maple does it too:

> with(Groebner);
> Basis({(t^2+1)^3*x-2*t*(3*t^4+50*t^2-33), (t^2+1)^3*y-2*(7*t^6-60*t^4+15*t^2+2)}, tdeg(t, x, y));
Related Question