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

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

### Reparameterization

Let $\phi: \closedint c d \to \closedint a b$ be a bijective differentiable strictly increasing real function.

Let $\sigma : \closedint c d \to \C$ be defined by:

$\sigma = \gamma \circ \phi$

Then $\sigma$ is 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.