Reparameterization of Directed Smooth Curve Preserves Image

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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 $\operatorname{Im} \left({\gamma}\right) = \operatorname{Im} \left({\sigma}\right)$, where $\operatorname{Im} \left({\gamma}\right)$ denotes the image of $\gamma$.


Proof

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

From Surjection by Restriction of Codomain, it follows that there exists a function $\tilde{\gamma}: \left[{a \,.\,.\, b}\right] \to \operatorname{Im} \left({\gamma}\right)$ that is the surjective restriction of $\gamma$ to $\operatorname{Im} \left({\gamma}\right)$.

Put $\tilde{\sigma} := \tilde{\gamma} \circ \phi : \left[{c \,.\,.\, d}\right] \to \operatorname{Im} \left({\gamma}\right)$.

From Composite of Surjections is Surjection, it follows that $\tilde{\sigma}$ is surjective.

Then:

$\operatorname{Im} \left({\tilde{\sigma} }\right) = \operatorname{Im} \left({\gamma}\right)$

Now:

$\sigma \left({t}\right) = \tilde{\sigma} \left({t}\right)$ for all $t \in \left[{c \,.\,.\, d}\right] = \operatorname{Dom} \left({\sigma}\right) = \operatorname{Dom} \left({\tilde{\sigma} }\right)$

where $\operatorname{Dom} \left({\sigma}\right)$ denotes the domain of $\sigma$.

It follows that:

$\operatorname{Im} \left({\sigma}\right) = \operatorname{Im} \left({\tilde{\sigma} }\right)$

Hence:

$\operatorname{Im} \left({\tilde{\gamma} }\right) = \operatorname{Im} \left({\sigma}\right)$

$\blacksquare$


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