# 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** if and only if:

- $(1): \quad$ 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:

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

Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

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

$C$ is a **simple contour** if and only if:

- $(1): \quad$ 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 $\gamma_i \left({t_1}\right) \ne \gamma_j \left({t_2}\right)$.

- $(2): \quad$ 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 $\gamma_k \left({t}\right) \ne \gamma_n \left({b_n}\right)$.

## Also see

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

## Notes

Note that a **simple contour** may be a closed contour, so $\rho_1 \left({a_1}\right) = \rho_n \left({b_n}\right)$.