# Definition:Directed Smooth Curve/Endpoints

## Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a directed smooth curve in $\R^n$.

Let $C$ be parameterized by a smooth path $\rho: \left[{a \,.\,.\, b}\right] \to \C$.

Then:

$\rho \left({a}\right)$ is the start point of $C$
$\rho \left({b}\right)$ is the end point of $C$.

Collectively, $\rho \left({a}\right)$ and $\rho \left({b}\right)$ are known as the endpoints of $\rho$.

### Complex Plane

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

Let $C$ be a directed smooth curve in the complex plane $\C$.

Let $C$ be parameterized by a smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$.

Then:

$\gamma \left({a}\right)$ is the start point of $C$
$\gamma \left({b}\right)$ is the end point of $C$.

Collectively, $\gamma \left({a}\right)$ and $\gamma \left({b}\right)$ are known as the endpoints of $\rho$.