# Definition:Directed Smooth Curve/Parameterization/Complex Plane

## Definition

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.