Definition:Directed Smooth Curve/Endpoints/Complex Plane

Definition
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 $\rho$.

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 $\gamma$.