Definition:Directed Smooth Curve/Parameterization

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Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a directed smooth curve in $\R^n$.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \C$ 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 $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ 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 $C$.

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.