Hello!

I've got a question about the definition of plane curves as equivalence classes under the relation of reparametrization.

It is said that whenever two plane curves have different trace, i.e. their graphical representation on the plane differs, the curves belong to distinct equivalence classes.

The converse is not true, however. The two curves $\displaystyle c_1:[0,2\pi] \longrightarrow \mathbb{R}^2$ and $\displaystyle c_2:[0,4\pi] \longrightarrow \mathbb{R}^2$ defined by $\displaystyle c_i(t)=(\cos t, \sin t)$ are different in the above sense, in spite of the fact that they have the same trace. This is odd, since I can reparametrize one to the other by means of the map $\displaystyle \varphi(t)=2t$.

Is it just because the cirve is closed and periodic? How do we actually unambiguously distinguish closed curves/periodic curves?