Definition:Directed Smooth Curve/Parameterization/Complex Plane

Definition
Let $C$ be a directed smooth curve.

A smooth path $\gamma : \left[{a \,.\,.\, b}\right] \to \C$ is called a parameterization of $C$ iff $\gamma$ is a member of the equivalence class that constitutes $C$. That is, if the smooth paths $\gamma$ and $\sigma: \left[{c \,.\,.\, d}\right] \to \C$ both are members of $C$, then there exists a bijective differentiable strictly increasing function $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ such that $\sigma = \gamma \circ \phi$.

From Directed Smooth Curve Relation is Equivalence, it follows that this defines an equivalence relation on the set of all smooth paths: $\left\{ { \gamma: I \to \C }\, \middle\vert \,{ \text{$I$ is a closed real interval, $\gamma$ is a smooth path} }\right\}$.

If another smooth path $\sigma$ is also a member of $C$, then $\sigma$ may be called a reparameterization of $C$.