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 $C$.

If another smooth path $\sigma : \left[{c \,.\,.\, d}\right] \to \C$ is also a member of $C$, then $\sigma$ may be called a reparameterization of $C$. That is, there exists a bijective differentiable strictly increasing function $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ such that $\sigma = \gamma \circ \phi$.