We want to study a parametric curve $f : \mathbb{R} \to \mathbb{R}^2, \ t\mapsto (x(t),y(t))$.
To proceed, we use the cartesian equation of a line $y=ax+b$ with $a\in \mathbb{R},b\in \mathbb{R}^{*}$ and determine the points of intersection with the parametric curve changing the value of $b$. Then it enables to have an idea of the drawing of the parametric curve.
Is this an established method of analyzing curves? If so, what is its name?