I have a small confusion about curves. Given a non-simple parametric curve $\alpha : I \mapsto \mathbb{R}^{2}$.
How does the tangent vector $T(s)$ and the tangent line $\{\alpha(s) + t T(s) : t \in \mathbb{R} \}$ look like at this point of intersection?
Everytime a particle traces this point, there will be a different tangent vector pointing in the direction motion and it will have more than one tangent line passing through it in general. Is this correct? Can we say the same thing about any $\mathbb{R}^{n}$ curve?
I feel like this could be problematic, but not sure why.
There is no single tangent line at the point in question. (In fancy language, this is an immersed $1$-dimensional submanifold, but not a submanifold.) You will get, in this example, one tangent line for each of the two values of $s$, say $s_1$ and $s_2$, with $\alpha(s_1)=\alpha(s_2)=(0,0)$. There is nothing special here about $\Bbb R^2$; the same phenomenon can occur in higher dimensions (and for higher-dimensional objects than curves, too).