# Definition:Directed Smooth Curve/Parameterization/Complex Plane

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## Definition

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 $C$.

That is, if the smooth paths $\gamma$ and $\sigma: \left[{c \,.\,.\, d}\right] \to \C$ both are elements of $C$, then $\sigma = \gamma \circ \phi$.

Here $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ is a bijective differentiable strictly increasing function.

From Directed Smooth Curve Relation is Equivalence, it follows that this defines an equivalence relation on $G$, the set of all smooth paths.

Here $G$ is defined as $G = \left\{ {\gamma: I \to \C} \mid {\text{$I$ is a closed real interval, $\gamma$ is a smooth path} }\right\}$.

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**.

## Sources

- 2001: Christian Berg:
*Kompleks funktionsteori*$\S 2.2$