  In planar ${\cal N}=4$ supersymmetric Yang-Mills theory we have studied
supersymmetric Wilson loops composed of a large number of light-like segments,
i.e., null zig-zags. These contours oscillate around smooth underlying
spacelike paths. At one-loop in perturbation theory we have compared the finite
part of the expectation value of null zig-zags to the finite part of the
expectation value of non-scalar-coupled Wilson loops whose contours are the
underlying smooth spacelike paths. In arXiv:0710.1060 [hep-th] it was argued
that these quantities are equal for the case of a rectangular Wilson loop. Here
we present a modest extension of this result to zig-zags of circular shape and
zig-zags following non-parallel, disconnected line segments and show
analytically that the one-loop finite part is indeed that given by the smooth
spacelike Wilson loop without coupling to scalars which the zig-zag contour
approximates. We make some comments regarding the generalization to arbitrary
shapes.
