I'm confused about the concept of equivalent parametric curves. Based on my understanding, two parametric curves, $\phi$ and $\psi$, are equivalent, if there is a strictly monotonically increasing function $g$ such that $\psi(g(t)) = \phi(t)$. Intuitively speaking, the two curves must have the same direction and the same image, and they "travel" through their image for the same number of times.
(1) Why does $g$ have to be monotonically increasing? Why can't it be decreasing?
(2) They are allowed to have different speeds, correct?
(3) Are these two equivalent: $\phi(\theta) = (cos(\theta),sin(\theta))$ and $\psi(\theta) = (cos(2\theta),sin(2\theta))$, where the domain of $\phi$ is $[0,2\pi]$, and the domain of $\psi$ is $[0,\pi]$?