Take the following parametric equation of an implicit curve as an example:
$$ \left\{\quad \begin{array}{rl} x=& \dfrac{27}{14} \sin 2 t+\dfrac{15}{14} \sin 3 t \\ y=& \dfrac{27}{14} \cos 2 t-\dfrac{15}{14} \cos 3 t \\ \end{array} \right. $$
which has five distinct intersection points with itself:
How to determine the five intersection points in $t$? What kind of equation systems should be used in order to avoid an infinite dimensional solution set?
The method to find the intersections points consists to consider two values of the parameter $t$, namely $t_1$ and $t_2$. Since the point is common, the corresponding coordinates are respectively equal $x_1=x_2$ and $y_1=y_2$. This leads to a system of equations which solving leads a set of values of $t$. Then, keeping only the real roots consistent with the initial equations, one obtains the five points of intersection. The coordinates of the points are given in the table below.
The exact results are :
First point : $$\begin{bmatrix} x=0 \\ y=-\frac{6}{25}(\sqrt{181}-9) \end{bmatrix}$$ Second and fifth points : $$\begin{bmatrix} x=\pm\frac{3}{50}\sqrt{10-2\sqrt{5}} (\sqrt{181}-9) \\ y=\frac{3}{50}(\sqrt{5}+1) (\sqrt{181}-9) \end{bmatrix}$$ Third and fourth points : $$\begin{bmatrix} x=\pm\frac{3}{50}\sqrt{10+2\sqrt{5}} (\sqrt{181}-9) \\ y=-\frac{3}{50}(\sqrt{5}-1) (\sqrt{181}-9) \end{bmatrix}$$