A slight modification of Rohits solution might simply be:
Show[img, ListLinePlot[adat, AspectRatio -> 1, PlotStyle -> Red, InterpolationOrder -> 3]]
So for hitting intermediate points you basically need interpolation. If you need some function describing this spiral, you can do it like so:
shell[t_] = Evaluate[Through[(ListInterpolation[#, {0, 1}, InterpolationOrder -> 3] & /@ Transpose[adat])[t]]];
ParametricPlot[shell[t], {t, 0, 1}]
