Showing if two regular parametrized curves are equivalent

Question

How do I know if the following two regular parametrized curves, $c_1$ and $c_2$, are equivalent?

(1) $c_1$ : [0, 2$\pi$] $\to$ $\mathbb{R}^2$ : $t$ $\mapsto$ (cos $t$, sin $t$)

(2) $c_2$ : [0, 2$\pi$] $\to$ $\mathbb{R}^2$ : $t$ $\mapsto$ (cos 2$t$, sin 2$t$)

I know that two curves are equivalent if one is a reparametrization of the other. What exactly does that mean and how do I show that one is a reparametrization of the other?

Answer 1

Let $\tilde{\gamma}:\tilde{I} \to \mathbb{R}^n$ and $\gamma:I \to \mathbb{R}^n$ be smooth maps. We say that $\tilde{\gamma}$ is a reparametrization of $\gamma$ if there exists a diffeomorphism $\phi: \tilde{I} \to I$ such that $(\gamma \circ \phi)(\tilde{t}) = \tilde{\gamma}(\tilde{t})$.

Answer 2

Intuitively, $c_1$ goes around the circle once, but $c_2$ goes around the circle twice. They have different 'paths', so they are not equivalent.

Rigorously, length of a curve is preserved under reparametrization. Note that $c_1$ and $c_2$ have different lengths.