Definition:Directed Smooth Curve/Endpoints/Complex Plane

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

Then $\gamma\left({a}\right)$ is called the start point of $C$, and $\gamma\left({b}\right)$ is called the endpoint of $C$.

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 $\gamma$.