Definition:Directed Smooth Curve

Definition
Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ be a smooth path.

Then the directed smooth curve with parameterization $\gamma$ is defined as an equivalence class of smooth paths.

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 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 $\gamma$.

Also known as
A directed smooth curve is called an oriented smooth curve, a smooth curve with orientation or simply a curve in many texts.

Linguistic Note
There are four valid spellings of parameterization: parameterization and parametrization (both US spellings), parameterisation and parametrisation (both UK spellings).