\begin{definition}[Definition:Directed Smooth Curve/Complex Plane]
Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ be a smooth path in $\C$.
The '''directed smooth curve''' with '''parameterization''' $\gamma$ is defined as an equivalence class of smooth paths as follows:
A smooth path $\sigma: \left[{ a \,.\,.\, b }\right] \to \C$ belongs to the equivalence class of $\gamma$ {{iff}}:
: there exists a bijective differentiable strictly increasing real function:
:: $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$
: such that $\sigma = \gamma \circ \phi$.
It follows from Directed Smooth Curve Relation is Equivalence and Fundamental Theorem on Equivalence Relations that this does in fact define an equivalence class.
If a '''directed smooth curve''' is only defined by a smooth path $\gamma$, then it is often denoted with the same symbol $\gamma$.
\end{definition}