  The Dirichlet Laplacian in a curved three-dimensional tube built along a
spatial (bounded or unbounded) curve is investigated in the limit when the
uniform cross-section of the tube diminishes. Both deformations due to bending
and twisting of the tube are considered. We show that the Laplacian converges
in a norm-resolvent sense to the well known one-dimensional Schroedinger
operator whose potential is expressed in terms of the curvature of the
reference curve, the twisting angle and a constant measuring the asymmetry of
the cross-section. Contrary to previous results, we allow the reference curves
to have non-continuous and possibly vanishing curvature. For such curves, the
distinguished Frenet frame standardly used to define the tube need not exist
and, moreover, the known approaches to prove the result for unbounded tubes do
not work. Our main ideas how to establish the norm-resolvent convergence under
the minimal regularity assumptions are to use an alternative frame defined by a
parallel transport along the curve and a refined smoothing of the curvature via
the Steklov approximation.
