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:

Also see

 * Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, where it is shown that the definitions are independent of the choice of parameterization $\rho$.