Detection of self intersection point of curve

Question

What numerical procedure is be adopted to detect self-intersecting parametrized points $ [x(t), y(t) ] $ in $ \mathbb R^2 $ ?

Observation : @ roots ( t= 2, t=-1 ) parabola has double value with respect to cubic. How to build an algorithm ?

Answer 1

I think the way to find out is that: $\exists t_1 \neq t_2: (x(t_1),y(t_1)) = (x(t_2),y(t_2))$. For if $r(t) = (x(t),y(t))$ is a continuously differentiable function on $\mathbb{R}$, then if $r'(t) = 0$ for some $t$ then you have a self intersection point.

Example: $r(t) = (t^2-t, t^3-t), t \in \mathbb{R}$.

We have $r(0) = (0,0) = r(1)$.