\begin{definition}[Definition:Reparametrization of Admissible Curve]
Let $M$ be a smooth manifold.
Let $\closedint a b, \closedint c d$ be closed real intervals.
Let $\gamma : \closedint a b \to M$ be an admissible curve.
Let $\phi : \closedint c d \to \closedint a b$ be a homeomorphism.
Let $\tuple {c_0, \ldots, c_k}$ be a finite subdivision of $\closedint c d$.
Suppose $\tuple {c_0, \ldots, c_k}$ is such that for all $i \in \N : 1 \le i \le k$ the restriction of $\phi$ to each subinterval $\closedint {c_{i - 1}} {c_i}$ is a diffeomorphism onto its image.
Then $\tilde \gamma := \gamma \circ \phi$ is called the '''reparametrization of the admissible curve $\gamma$''', where $\circ$ denotes the composition of mappings $\gamma$ and $\phi$.
\end{definition}