Self-intersection of vector valued function

Question

A vector valued function $r(t)$ has the following coordinates: $$x = 4\cos\left(\frac12t\right)+2\cos(2t)+\cos(4t)\\ y = 4\sin\left(\frac12t\right)+2\sin(2t)+\sin(4t)$$

I have to find the $t$-values where the curve crosses/intersects itself.

Using Maple I've not been able to find any solutions. I've tried a lot of combinations of setting the $x$- and $y$-coordinates equal each other, with no luck.

Answer 1

You don't want to set the $x$ and $y$ coordinates equal to each other. For example, the system

$ \begin{cases} x(t) = t\\ y(t) = t \end{cases} $

has $x=y$ everywhere but it never crosses itself.

Each of $x$ and $y$ in your system has period of $4\pi$, so it will cross itself.