= Prescribed-speed time parametrization of a curve
For a regular <parametric curve> $F(u)$ traversed in increasing $u$ with positive <speed> $v(F(u))$, its travel time is
$$
t(u)-t(u_0)=\int_{u_0}^{u}\frac{\|F'(\xi)\|}{v(F(\xi))}\,d\xi.
$$
Invert this strictly increasing function to sample at equal time intervals, or integrate $\dot u=v(F(u))/\|F'(u)\|$. 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.
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