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 $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k}{b_k} \to \C$ for all $k \in \set{ 1, \ldots, n}$.

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

The start point of $C$ is $\map {\gamma_1}{a_1}$.

The end point of $C$ is $\map {\gamma_n}{b_n}$.

Collectively, $\map {\gamma_1}{a_1}$ and $\map {\gamma_n}{b_n}$ 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 $C_1, \ldots, C_n$.