Definition:Contour/Simple

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

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

$C$ is a simple contour :


 * $(1): \quad$ For all $i, j \in \left\{ {1, \ldots, n}\right\}, t_1 \in \left[{a_i \,.\,.\, b_i}\right), t_2 \in \left[{a_j \,.\,.\, b_j}\right)$ with $t_1 \ne t_2$, we have $\rho_i \left({t_1}\right) \ne \rho_j \left({t_2}\right)$


 * $(2): \quad$ For all $k \in \left\{ {1, \ldots, n}\right\}, t \in \left[{a_k \,.\,.\, b_k}\right)$ where either $k \ne 1$ or $t \ne a_1$, we have $\rho_k \left({t}\right) \ne \rho_n \left({b_n}\right)$.

Thus a simple contour is a contour that does not intersect itself.

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 the parameterizations of $C_1, \ldots, C_n$.