Definition:Parameterization
Definition
Parameterization of Directed Smooth Curve
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 $C$.
Parameterization of Contour
Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$.
Let $C_i$ be parameterized by the smooth path $\rho_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$.
Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.
The parameterization of $C$ is defined as the function $\rho: \closedint {a_1} {c_n} \to \R^n$ with:
- $\map {\rho \restriction_{\closedint {c_i} {c_{i + 1} } } } t = \map {\rho_i} t$
where:
- $\ds c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \set {0, \ldots, n}$
- $\rho \restriction_{\closedint {c_i} {c_{i + 1} } }$ denotes the restriction of $\rho$ to $\closedint {c_i} {c_{i + 1} }$.
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.