Arc-length parametrization

For a regular curve, $s(t)={\int }_{t_0}^t|{\alpha }^{\prime }(r)|dr$ has positive derivative and hence a smooth inverse. Reparametrization by $s$ gives unit speed.

For a regular parametric curve $F(u)$ traversed in increasing $u$ with positive speed $v(F(u))$, its travel time isInvert this strictly increasing function to sample at equal time intervals, or integrate $\dot{u}=v(F(u))/\Vert F^{\prime }(u)\Vert$. A zero of the speed requires separate analysis: whether travel time diverges depends on the integrability of this reciprocal-speed expression, and the scalar speed alone does not determine the direction after a turning event.