This problem is generated from another Green's theorem related question of mine. And here is a forward of the same problem in math.stackexchange.
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:
- http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.204.9611
- 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:
$$\begin{cases} x=& \dfrac{2 t \left(3 t^4+50 t^2-33\right)}{\left(t^2+1\right)^3} \\ y=& \dfrac{2 \left(7 t^6-60 t^4+15 t^2+2\right)}{\left(t^2+1\right)^3} \\ \end{cases}$$
How can I convert it into implicit form :$F(x,y)=0$? My difficulty mainly lies in how to implement it via my familiar computer algebra systems, i.e., Mathematica, or Maple.
The answer by Macaulay2 is definitely acceptable, however, for a beginner like me, there are less documents on the commands than Maple or Mathematica.
GroebnerBasis[Thread[{x, y} == (* equations *)], {x, y}, t]
$\endgroup$GroebnerBasis
gain support for F4/F5 methods? $\endgroup$