I am interested in a particular method of studying a parametric curve that I don't fully comprehend, it's steps are:
Study of the parametric curve: $r(t)=(x(t),y(t))$
Reducing the domain
1. Period: find the period $T$ of the function, and set the domain as $D=[\alpha, \alpha +T]$
2. Symmetry: find the axis of symmetries by using a function of the parameter $t$. For example, let $\phi(t)$ be that function, then if $x(\phi(t))=x(t)$ and $y(\phi(t))=-y(t)$ then the $x$ axis is a axis of symmetry.
3. Table of variation: study the signs of the derivatives $x'(t)$ and $y'(t)$ then the variations and limits of $x(t)$ and $y(t)$ and finally the slope of the tangent.
4. Infinite branches: limits of the curve
Can someone give a full and detailed insight of this method to study parametric curves? Why did we choose these steps? How to fully study a parametric curve using this method? A full explanation is really appreciated.