= Reparametrization
A regular reparametrization of a <smooth curve> $\gamma:I\to M$ is $\widetilde\gamma=\gamma\circ\phi$, where $\phi:J\to I$ is a <diffeomorphism> of parameter <intervals>. The <chain rule> gives
$$
\widetilde\gamma'(u)=\gamma'(\phi(u))\phi'(u).
$$
It preserves the image and takes a <regular curve> to another <regular curve>, while multiplying <speed> by $|\phi'(u)|$. The sign of $\phi'$ determines whether the direction of traversal is preserved or reversed.
A merely smooth parameter substitution can destroy regularity: $\gamma(t)=(t,0)$ and $\phi(u)=u^3$ give $\widetilde\gamma'(0)=0$. The inverse cube-root <function> is not smooth there, so this is not a regular reparametrization. By contrast, <arc-length parametrization> uses a substitution with positive <derivative> and produces <speed> one.
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