As the Integral Curves technique develops, I am testing it on a various procedural modeling and animation problems. In this example, this technique was used for modeling of both the head and the wire of a safety pin, and then for animation of the wire. (This safety pin has a Russian-style head).
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An Integral Curve can bee seen as a path of a horizontally flying aircraft. The Azimuth is the deviation of the flying direction from the Polar Axis. The Elevation is the deviation of the path from the local horizontal plane. In this model, the rate of azimuth change with respect of traversed path is the Curvature (or Azimuth Differential). Thus, the path can be constructed by integrating azimuth and elevation, and the azimuth in turn by integrating curvature.
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The explanatory video.