Definition:Directed Smooth Curve
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 $\rho: \closedint a b \to \C$ 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 $\gamma : \closedint a b \to \C$ 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 $\sigma = \gamma \circ \phi$.
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.