Reversed Directed Smooth Curve is Directed Smooth Curve

Theorem
Let $C$ be a directed smooth curve in $\C$.

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

Define $\psi: \left[{a \,.\,.\, b}\right] \to \left[{a \,.\,.\, b}\right]$ by $\psi \left({t}\right) = a + b - t$.

Define $\rho: \left[{a \,.\,.\, b}\right] \to \C$ by $\rho = \gamma \circ \psi$.

Then $\rho$ is a smooth path which parameterizes a directed smooth curve $-C$.

The directed smooth curve $-C$ is independent of the parameterization $\gamma$.

Proof
First, we prove that $\rho$ is a smooth path:

As $\gamma'$ is continuously differentiable, and $\psi$ is continuous, it follows that $\rho'$ is continuous.

Then $\rho$ is a parameterization of a directed smooth curve $-C$.

Next, we prove that the definition of $-C$ is independent of the parameterization $\gamma$.

Suppose $\sigma$ is another parameterization of $C$:
 * $\sigma = \gamma \circ \phi$

Here $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ is a bijective differentiable strictly increasing function.

Define $\tilde \psi: \left[{c \,.\,.\, d}\right] \to \left[{c \,.\,.\, d}\right]$ by $\tilde \psi \left({t}\right) = c + d - t$.

Define $\tilde \rho: \left[{c \,.\,.\, d}\right] \to \C$ by $\tilde \rho = \sigma \circ \tilde{\psi}$.

We now prove that $\rho$ and $\tilde \rho$ both are parameterizations of the same directed smooth curve $-C$.

Both $\psi$ and $\tilde \psi$ are bijective with $\psi^{-1} = \psi$ and $\tilde \psi^{-1} = \tilde \psi$.

Define $\tilde \phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ by $\tilde \phi = \psi^{-1} \circ \phi \circ \tilde \psi$.

From Composite of Bijections is Bijection, it follows that $\tilde \phi$ is bijective.

From Derivative of Composite Function, it follows that $\tilde \phi$ is differentiable with

From Derivative of Monotone Function, it follows that $\tilde \phi$ is strictly increasing.

As:


 * $\rho \circ \tilde \phi = \gamma \circ \psi \circ \psi^{-1} \circ \phi \circ \tilde \psi = \gamma \circ \phi \circ \tilde \psi = \sigma \circ \tilde \phi = \tilde \rho$

it follows that $\rho$ and $\tilde{\rho}$ are parameterizations of $-C$.