Cycloids

A cycloid is the curve traced by a fixed point on a circle as the circle rolls along a straight line without slipping. If the rolling circle has radius r and the tracing point is at a distance d from the center, the curve can be described parametrically by x = r t − d sin t and y = r − d cos t, with t representing the angle of rotation.

The ordinary cycloid corresponds to a point on the circumference, where d = r. This gives x = r(t − sin t) and y = r(1 − cos t). The baseline of the curve is the line along which the circle rolls.

Variants of the cycloid include the curtate cycloid (0 < d < r) and the prolate cycloid (d > r). The curtate cycloid has no cusps and stays entirely above the baseline, while the prolate cycloid can dip below the baseline and may form loops.

Key geometric features include cusps and arch length. For the ordinary cycloid, cusps occur at t = 2πk, where the tracing point touches the baseline. The length of one arch of the standard cycloid is 8r.

The cycloid has important mathematical properties. It is the tautochrone, meaning the time for a bead sliding under gravity to reach the bottom point is the same regardless of the starting point on the curve. It is also the brachistochrone, the curve of quickest descent between two points under uniform gravity.

History and applications. The cycloid was studied extensively in the 17th century, with foundational work by Huygens on the tautochrone and brachistochrone problems. Beyond theory, cycloidal curves appear in engineering contexts such as cycloidal drives and in architectural and optical designs.