A horseshoe curve is a class of climbing curve in a roadbed which reverses turn direction (inflection) twice on either side of a single tight curve that varies through an angle of about 180 degrees or more. Such curves are more commonly found in a railway line but are also used in roads. The characteristic U shape, or even slight balloon shape, of such a curve resembles a horseshoe. On roadways, particularly tight versions of such curves are typically called hairpin turns. Theory A horseshoe curve is a means to lengthen an ascending or descending grade and thereby reduce the maximum gradient. Grade or gradient is defined as the rise divided by the run (length) or distance, so in principle such curves add to length for the same altitude gain, just as would a climbing spiral around one or more peaks, or a climbing traverse (cutting) wrapping around an end of a ridge. If the straight route between two points is too steep to climb, a more circuitous route will increase the distance traveled, allowing the difference in altitude to be averaged over a longer track (or road) length. Unlike a spiral, a horseshoe curve does not involve the track crossing over itself, and the full horseshoe involves both relatively straight sections, curve deflections in both directions and tightly curved segment; while a spiral generally has a more uniform curvature. Obviously, a horseshoe also gives rise to a severe change in direction requiring another corrective curve to regain displacement in the overall direction of travel, while a spiral generally does not. A horseshoe curve is sometimes used where the route bridges a deep gully. Deviating from a straight-line route along the edge of the gully may allow it to be crossed at a better location. Horseshoe curves are common on railway lines in steeply graded or hilly country, where means must be found to achieve acceptable grades and minimize construction costs. As with spirals, the main limitation in laying out a horseshoe is keeping its radius as large as possible, as sharp curves limit train speed, and through increased friction, are harder on rails, requiring more frequent replacement of outer tracks. Examples Europe Germany The Rhein-Ruhr S-Bahn in Germany has a horseshoe curve in Neviges, Velbert on the route between Essen and Wuppertal, known as the Prince William railway. Norway The Dovre Line, the main line of the Norwegian railway