Logarithmic conical helix ($x(s)=(\frac{s}{2}\cos \log s,\frac{s}{2}\sin \log s,s/\sqrt{2})$)

The displayed space curve, $s>0$, lies on a cone and is parametrized by arc length. It has a constant tangent angle with its axis, while the turn spacing shrinks geometrically toward the limiting apex. Its curvature is $1/(\sqrt{2}s)$: differentiate its unit tangent with respect to $s$. Infinitely many turns accumulate at the apex with finite arc length; the apex itself is not a regular point of the curve.