Reversed Directed Smooth Curve is Directed Smooth Curve

Theorem
Let $C$ be a directed smooth curve.

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 $\psi$ are 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 $C$, so $\sigma = \gamma \circ \phi$.

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 $-C$.