The shapes of spiral arms in the S$^4$G survey and their connection with
  stellar bars
  Spiral galaxies are common in the local Universe, but their formation,
evolution, and interplay with bars remain poorly understood. We use a sample of
391 nearby galaxies from the S$^4$G survey to characterise the winding angle
and amplitude of spiral arms as a function of disc properties, such as bar
strength, in all kinds of spirals (grand-design, multi-armed, and flocculent).
We derive global pitch angles in 3.6 $\mu$m de-projected images from i) average
measurements of individual logarithmic spiral segments, and ii) for a subsample
of 32 galaxies, from 2-D Fourier analyses. The strength of spirals is
quantified from the tangential-to-radial force ratio and from the normalised
$m=2$ Fourier density amplitudes. In galaxies with more than one measured
logarithmic segment, the spiral pitch angle varies on average by $\sim
10^{\circ}$ between segments, but by up to $\gtrsim 15-20^{\circ}$. The
distribution of the global pitch angle versus Hubble type ($T$) is very similar
for barred and non-barred galaxies when $1 \lesssim T \lesssim 5$. Most spiral
galaxies ($>90\%$) are barred for $T>5$. The pitch angle is not correlated with
bar strength, and only weakly with spiral strength. The amplitude of spirals is
correlated with bar strength (and less tightly, with bar length) for all types
of spirals. The mean pitch angle is hardly correlated with the mass of the
supermassive black hole (estimated from central stellar velocity dispersion),
with central stellar mass concentration, or with shear, questioning previous
results in the literature using smaller samples. We do not find observational
evidence that spiral arms are driven by stellar bars or by invariant manifolds.
Most likely, disks that are prone to the development of strong bars are also
reactive to the formation of prominent spirals, explaining the observed
coupling between bar and spiral amplitudes (Abridged).
