Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints

Theorem
Let $\left[{a \,.\,.\, b}\right]$ and $\left[{c\,.\,.\, d}\right]$ be closed real intervals.

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

Let $C$ be a directed smooth curve with parameterization $\gamma$.

Suppose that $\sigma : \left[{c \,.\,.\, d}\right] \to \C$ is a reparameterization of $C$.

Then the start points and end points of $\gamma$ and $\sigma$ are identical:


 * $\gamma \left({a}\right) = \sigma \left({c}\right)$


 * $\gamma \left({b}\right) = \sigma \left({d}\right)$

Proof
By definition of reparameterization, there exists a bijective differentiable strictly increasing function $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ such that $\sigma = \gamma \circ \phi$.

As $\phi^{-1} \left({a}\right) \in \left[{c \,.\,.\, d}\right]$, it follows that $c \le \phi^{-1} \left({a}\right)$.

As $\phi$ is strictly increasing, we have $\phi \left({c}\right) \le \phi \left({\phi^{-1} \left({a}\right) }\right) = a$.

As $\phi \left({c}\right) \in \left[{a \,.\,.\, b}\right]$, it follows that $\phi \left({c}\right) = a$.

Hence:


 * $\sigma \left({c}\right) = \gamma \circ \phi \left({c}\right) = \gamma \left({a}\right)$

As $\phi^{-1} \left({b}\right) \in \left[{c \,.\,.\, d}\right]$, it follows that $d \ge \phi^{-1} \left({b}\right)$.

As $\phi$ is strictly increasing, we have $\phi \left({d}\right) \ge \phi \left({\phi^{-1} \left({b}\right) }\right) = b$.

As $\phi \left({d}\right) \in \left[{a \,.\,.\, b}\right]$, it follows that $\phi \left({d}\right) = b$.

Hence:


 * $\sigma \left({d}\right) = \gamma \circ \phi \left({d}\right) = \gamma \left({b}\right)$