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.