# Definition:Directed Smooth Curve/Endpoints

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## 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 $\rho$.

## 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 $\rho$.