# Definition:Contour/Endpoints

## 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 defined by the finite sequence $C_1, \ldots, C_n$.

The start point of $C$ is $\rho_1 \left({a_1}\right)$.

The end point of $C$ is $\rho_n \left({b_n}\right)$.

Collectively, $\rho_1 \left({a_1}\right)$ and $\rho_n \left({b_n}\right)$ are referred to as the endpoints of $C$.

### 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 $\C$.

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 $C_1, \ldots, C_n$.

The start point of $C$ is $\gamma_1 \left({a_1}\right)$.

The end point of $C$ is $\gamma_n \left({b_n}\right)$.

Collectively, $\gamma_1 \left({a_1}\right)$ and $\gamma_n \left({b_n}\right)$ are referred to as the endpoints of $C$.

## Also see

From Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, it follows that this definition is independent of the parameterizations of $C_1, \ldots, C_n$.