# Definition:Directed Smooth Curve/Parameterization

## Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a directed smooth curve in $\R^n$.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\R^n$.

Then $\rho$ is a **parameterization** of $C$ if and only if $\rho$ is an element of the equivalence class that constitutes $C$.

### Complex Plane

The definition carries over to the complex plane, in which context it is usually applied:

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

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

Then $\gamma$ is a **parameterization** of $C$ if and only if $\gamma$ is an element of the equivalence class that constitutes $C$.

If another smooth path $\sigma$ is also an element of $C$, then $\sigma$ may be called a **reparameterization** of $C$.

## Linguistic Note

There are four valid spellings of this term:

**parameterization**and**parametrization**(both US spellings)**parameterisation**and**parametrisation**(both UK spellings).

This site prefers the more modern US spelling **parameterization**.