I would like to know if there is a way to determine mathematically (given some parameter) the equivalence of two curves given by different parameterizations.
For example, the two curves defined respectively by
$x_1 = \sin(t),\ y_1 = \cos(t)$ for $t\in[0,2\pi]$
$x_2 = \cos(2t),\ y_1 = \sin(2t)$ for $t\in[0,\pi]$
trace the same circle and are both solutions to the equation of the circle: $x^2+y^2 = 1$. However, I am wondering if there is a on operation that can be done on both curves to determine whether they trace the same implicit curve or not?