\begin{definition}[Definition:Directed Smooth Curve/Parameterization]
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 $C$ {{iff}} $\rho$ is an element of the equivalence class that constitutes $C$.
\end{definition}