\begin{definition}[Definition:Directed Smooth Curve/Parameterization/Complex Plane]
Let $C$ be a directed smooth curve in the complex plane $\C$.
Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\C$.
Then $\gamma$ is a '''parameterization''' of $C$ {{iff}} $\gamma$ is an element of the equivalence class that constitutes $C$.
That is, if the smooth paths $\gamma$ and $\sigma: \left[{c \,.\,.\, d}\right] \to \C$ both are elements of $C$, then $\sigma = \gamma \circ \phi$.
Here $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ is a bijective differentiable strictly increasing function.
From Directed Smooth Curve Relation is Equivalence, it follows that this defines an equivalence relation on $G$, the set of all smooth paths.
Here $G$ is defined as $G = \left\{ {\gamma: I \to \C} \mid {\text{$I$ is a closed real interval, $\gamma$ is a smooth path} }\right\}$.
If another smooth path $\sigma$ is also an element of $C$, then $\sigma$ may be called a '''reparameterization''' of $C$.
\end{definition}