When does a parametric curve on 2-D plane have self-loops?

Question

Consider a parametric curve on 2-D plane, denoted as $\Gamma = \{(x(t), y(t))\:|\: t\in [t_0, t_1]\subset \mathbb{R}\}$. Suppose $x(t), y(t)$ are polynomials of $t$. I'm wondering under what conditions on $x, y, t_0, t_1$ that $\Gamma$ has self-loops, i.e. $\mathbb{R}^2/ \Gamma$ is disconnected.

For instance, $\Gamma_1 = \{(t(t-1)(t-2), (t-1)(t-2)(t-3)) \:|\: t\in (0, 3)\}$ has a self-loop, while $\Gamma_2 = \{(t^2, -t^2)\:| \: t\in (-1, 1)\}$ does not.

EDIT: I'm more interested in, if there is any, sufficient and necessary conditions on $x, y, t_0, t_1$ that guarantees $\Gamma$ has self-loops.

Answer 1

Multiple points are solutions of $$\begin{cases}x(t)=x(t'),\\y(t)=y(t')\end{cases}$$ with $t\ne t'$.

They require at least one of the functions to be non-invertible. You can decompose the curve in continuous pieces where the derivatives keep a constant sign, and check for intersections of the pieces.

Answer 2

Suppose parametric form of a function is $(f(t),g(t))$. Then the curve will have a loop if and only if the curve intersects itself, that is, it acquires the same value of coordinates for two different values of $t$. Hence, for condition of loop, $$f(t_1)=f(t_2), g(t_1)=g(t_2)$$ must have a simultaneous solution.