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 $\S 2.2$