Definition:Directed Smooth Curve/Parameterization

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a directed smooth curve in $\R^n$.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\R^n$.

Then $\rho$ is a parameterization of $C$ $\rho$ is an element of the equivalence class that constitutes $C$.

Complex Plane
The definition carries over to the complex plane, in which context it is usually applied: