Travel time at reciprocal-curvature speed
= Travel time at reciprocal-curvature speed
{title2=$dt=\kappa\,ds$}
If a regular plane curve has positive <curvature> $\kappa$ and a particle moves with <speed> $v=1/\kappa$ in the chosen time units, its elapsed time is $\int\kappa\,ds$, the <total curvature> traversed. On a <logarithmic spiral> $r=ae^{b\theta}$, both $ds/d\theta$ and $1/\kappa$ equal $r\sqrt{1+b^2}$. Hence $dt=d\theta$ and every complete turn takes time $2\pi$, regardless of its radius.