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 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.

This equivalence class is denoted with the same symbol $\gamma$ as its parameterization.

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.

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