Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints

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Theorem

Let $\R^n$ be a real cartesian space of $n$ dimensions.

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

Let $\gamma: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$.

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 real function $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ such that $\sigma = \gamma \circ \phi$.

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

$c \le \phi^{-1} \left({a}\right)$

As $\phi$ is strictly increasing:

$\phi \left({c}\right) \le \phi \left({\phi^{-1} \left({a}\right) }\right) = a$

As $\phi \left({c}\right) \in \left[{a \,.\,.\, b}\right]$:

$\phi \left({c}\right) = a$

Hence:

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


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

$d \ge \phi^{-1} \left({b}\right)$

As $\phi$ is strictly increasing:

$\phi \left({d}\right) \ge \phi \left({\phi^{-1} \left({b}\right) }\right) = b$

As $\phi \left({d}\right) \in \left[{a \,.\,.\, b}\right]$:

$\phi \left({d}\right) = b$

Hence:

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

$\blacksquare$


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