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 $\S 2.2$