# Definition:Directed Smooth Curve/Complex Plane

## Definition

Let $\gamma : \closedint a b \to \C$ be a smooth path in $\C$.

The **directed smooth curve** with **parameterization** $\gamma$ is defined as an equivalence class of smooth paths as follows:

A smooth path $\sigma: \closedint c d \to \C$ belongs to the equivalence class of $\gamma$ if and only if:

- there exists a bijective differentiable strictly increasing real function:
- $\phi: \closedint c d \to \closedint a b$

- such that $\sigma = \gamma \circ \phi$.

It follows from Directed Smooth Curve Relation is Equivalence and Fundamental Theorem on Equivalence Relations that this does in fact define an equivalence class.

If a **directed smooth curve** is only defined by a smooth path $\gamma$, then it is often denoted with the same symbol $\gamma$.

### Parameterization

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

Let $\gamma: \closedint a b \to \C$ be a smooth path in $\C$.

Then $\gamma$ is a **parameterization** of $C$ if and only if $\gamma$ is a representative of the equivalence class that constitutes $C$.

### Reparameterization

Let $\phi: \closedint c d \to \closedint a b$ be a bijective differentiable strictly increasing real function.

Let $\sigma : \closedint c d \to \C$ be defined by:

- $\sigma = \gamma \circ \phi$

Then $\sigma$ is called a **reparameterization** of $C$.

### Endpoints

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 known as

A **directed smooth curve** is called an **oriented smooth curve**, a **smooth curve with orientation** or simply a **curve** in many texts.

## Sources

- 2001: Christian Berg:
*Kompleks funktionsteori*$\S 2.2$