  The method of Helsing and co-workers evaluates Laplace and related layer
potentials generated by a panel (composite) quadrature on a curve, efficiently
and with high-order accuracy for arbitrarily close targets. Since it exploits
complex analysis, its use has been restricted to two dimensions (2D). We first
explain its loss of accuracy as panels become curved, using a classical complex
approximation result of Walsh that can be interpreted as "electrostatic
shielding" of a Schwarz singularity. We then introduce a variant that swaps the
target singularity for one at its complexified parameter preimage; in the
latter space the panel is flat, hence the convergence rate can be much higher.
The preimage is found robustly by Newton iteration. This idea also enables, for
the first time, a near-singular quadrature for potentials generated by smooth
curves in 3D, building on recurrences of Tornberg-Gustavsson. We apply this to
accurate evaluation of the Stokes flow near to a curved filament in the slender
body approximation. Our 3D method is several times more efficient (both in
terms of kernel evaluations, and in speed in a C implementation) than the only
existing alternative, namely, adaptive integration.
