Definition:Directed Smooth Curve/Parameterization/Complex Plane

Definition
Let $C$ be a directed smooth curve in the complex plane $\C$.

Let $\gamma : \closedint a b \to \C$ be a smooth path in $\C$.

Then $\gamma$ is a parameterization of $C$ $\gamma$ is a representative of the equivalence class that constitutes $C$.

If another smooth path $\sigma : \closedint c d \to \C$ is a representative of $C$, then $\sigma$ is called a reparameterization of $C$.

Also see

 * Directed Smooth Curve Relation is Equivalence, where the equivalence relation of smooth paths is described.