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Use the prescribed-speed time parametrization of a curve. On a regular part traversed in increasing $u$, let $q(u)=\Vert F^{\prime }(u)\Vert >0$ and assume the prescribed speed $v(F(u))$ is positive. The chain rule for velocity givesIntegrate this scalar ordinary differential equation with an adaptive Runge-Kutta method, arranging output at the exact times $t_k=t_0+k\Delta t$. Evaluate $P_k=F(u(t_k))$. Internal integration steps may be shorter than $\Delta t$; output interpolation must meet the same error tolerance, so variable numerical steps do not create variable frame times.

An equivalent implementation precomputesby adaptive numerical integration. This function is strictly increasing. Bracket each $t_k$ in the table and invert using safeguarded Newton method or interval bisection; monotonicity makes each solution unique. Stop at $u=1$, with a shorter final time interval if the total travel time is not a multiple of $\Delta t$. This accounts for changes in both the curve's parameter speed and the physical speed, unlike equal increments in $u$.

If the specified speed vanishes, examine the reciprocal-speed integral at that event. It may diverge, giving asymptotic approach, or be integrable, giving arrival in finite time. A scalar speed does not determine whether motion reverses or how it resumes after stopping; such behavior needs the physical dynamics or a chosen continuation. The positive-speed formula applies between these events, with the direction changed if the traversal reverses.