Reparametrization

A regular reparametrization of a smooth curve $\gamma :I\to M$ is $\tilde{\gamma }=\gamma \circ \varphi$, where $\varphi :J\to I$ is a diffeomorphism of parameter intervals. The chain rule givesIt preserves the image and takes a regular curve to another regular curve, while multiplying speed by $|{\varphi }^{\prime }(u)|$. The sign of ${\varphi }^{\prime }$ determines whether the direction of traversal is preserved or reversed.

A merely smooth parameter substitution can destroy regularity: $\gamma (t)=(t,0)$ and $\varphi (u)=u^3$ give ${\tilde{\gamma }}^{\prime }(0)=0$. 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.