If I have a function $y = f(x)$, and I parameterize it so that I get two new functions $x(t)$ and $y(t)$, I'm curious about when I can say that two distinct parameterizations are equal, and in what sense they are equal. I'll give a trivial example:
Say $y = x, x \in [-1,1]$ is my original function. I might parameterize this function as:
$$x(t) = t$$
$$y(t) = t$$
$$t \in [-1,1]$$
This would, when plotted, yield the same points in the same order as the original function, and I think that the parameterization and the original function are equal in that regard. But let's now introduce a new parameterization:
$$x(t) = -cos(t)$$ $$y(t) = -cos(t)$$ $$t \in [0,\pi]$$
This new parameterization still produces the same points (and I think in the same order, but correct me if I'm wrong), and yet is qualitatively different from the original parameterization. In what sense can I say that these two parameterizations are equal, and in what sense are they not equal? How do I quantify that?