Definition:Directed Smooth Curve/Endpoints
< Definition:Directed Smooth Curve(Redirected from Definition:Endpoints of Directed Smooth Curve)
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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 $\rho$.