The cycloid (blue) and its generating circle (red).
Types of cycloid.
A cycloid is the shape defined by a fixed point on a wheel as it rolls. More precisely, it is the locus of a point on the rim of a circle rolling along a perfectly straight line. The curve resembles a succession of arches, with cusps separated by distances equal to the circumference of the circle.
The cycloid was named by Galileo in 1599. It is the solution to both the tautochrone problem and the brachistochrone problem. In 1634, the French mathematician Gilles de Roberval (1610–1675) showed that the area under a cycloid is three times the area of its generating circle. In 1658, the English architect Christopher Wren showed that the length of a cycloid is four times the diameter of its generating circle. But there was a lot of bickering and a lack of public sharing of information around this time that led to much duplication of effort, particularly over questions related to the cycloid. In fact the confusion was so bad that the curve was nicknamed the Helen of Geometers, and Jean Montucla referred to it as "la pomme de discorde." The cycloid is of architectural interest since it forms the strongest known arch.
As well as the ordinary cycloid there is the curtate cycloid, which is the path traced out by a point on the inside of a rolling circle, and the prolate cycloid, which is followed by a point on the outside of the circle. A prolate cycloid is traced out, for example, by points on the flange of the wheels of a locomotive, which extends below the top of the tracks. This leads to the surprising conclusion that even as the locomotive is moving forward there are always parts of its wheels that are going backward for a moment before moving forward again.