Definition:Contour/Image

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of 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 $i \in \left\{ {1, \ldots, n}\right\}$.

The image of $C$ is defined as:


 * $\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\rho_i}\right)$

where $\operatorname{Im} \left({\rho_i}\right)$ denotes the image of $\rho_i$.

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

Also see

 * Reparameterization of Directed Smooth Curve Preserves Image, from which it follows that this definition is independent of parameterizations of $C_1, \ldots, C_n$.