Definition:Contour/Parameterization

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Definition

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

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 $\closedint {c_i} {c_{i + 1} }$.


Note that this definition depends on the choice of parameterizations of $C_1, \ldots, C_n$.


Complex Plane

The definition carries over to the complex plane, in which context it is usually applied:


Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ 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 $\gamma: \closedint {a_1} {c_n} \to \C$ with:

$\map {\gamma \restriction_{\closedint {c_i} {c_{i + 1} } } } t = \map {\gamma_i} t$

where $\ds c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \set {0, \ldots, n}$.

Here, $\gamma \restriction_{\closedint {c_i} {c_{i + 1} } }$ denotes the restriction of $\gamma$ 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.


Sources