Definition:Reversed Directed Smooth Curve/Complex Plane

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

Let $C$ be parameterized by the smooth path $\gamma: \left[{a\,.\,.\,b}\right] \to \C$.

The reversed directed smooth curve of $C$ is denoted $-C$ and is defined as the directed smooth curve that is parameterized by:


 * $\rho: \left[{a\,.\,.\,b}\right] \to \C, \ \rho \left({t}\right) = \gamma \left({a + b - t}\right)$

From Reversed Directed Smooth Curve is Directed Smooth Curve, it follows that $-C$ is a directed smooth curve.

Also known as
A reversed directed smooth curve is called a reversed curve or a reciprocal curve in some texts.

Also denoted as
The reversed directed smooth curve of $C$ is denotes as $C^-$ in some texts.