Definition:Directed Smooth Curve

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Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$.


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

A smooth path $\sigma: \left[{a \,.\,.\, b}\right] \to \R^n$ belongs to the equivalence class of $\rho$ if and only if:

there exists a bijective differentiable strictly increasing real function:
$\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$
such that $\sigma = \rho \circ \phi$.


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 $\rho$, then it is often denoted with the same symbol $\rho$.


Parameterization

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a directed smooth curve in $\R^n$.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\R^n$.


Then $\rho$ is a parameterization of $C$ if and only if $\rho$ is an element of the equivalence class that constitutes $C$.


Endpoints

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 $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ 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 $\sigma: \left[{ a \,.\,.\, b }\right] \to \C$ belongs to the equivalence class of $\gamma$ if and only if:

there exists a bijective differentiable strictly increasing real function:
$\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$
such that $\sigma = \gamma \circ \phi$.


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.