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

### 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 $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.

The parameterization of $C$ is defined as the function $\rho: \left[{a_1 \,.\,.\, c_n}\right] \to \R^n$ with:

$\rho \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] } \left({t}\right) = \rho_i \left({t}\right)$

where:

$\displaystyle c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \left\{ {0, \ldots, n}\right\}$
$\rho \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] }$ denotes the restriction of $\rho$ to $\left[{c_i \,.\,.\, c_{i + 1} }\right]$.

## 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.