Дөңгелек домалағандағы нүктенің траекториясы

Curve traced by a point on a rolling circle

In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). In physics, when a charged particle at rest is put under a uniform electric and magnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid.

The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th century mathematicians, while Sarah Hart sees it named as such "because the properties of this curve are so beautiful". Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery speculated that such a simple curve must have been known to the ancients, citing similar work by Carpus of Antioch described by Iamblichus.

The involute of the cycloid has exactly the same shape as the cycloid it originates from. This can be visualized as the path traced by the tip of a wire initially lying on a half arch of the cycloid: as it unrolls while remaining tangent to the original cycloid, it describes a new cycloid (see also cycloidal pendulum and arc length).

Several curves are related to the cycloid. Trochoid: generalization of a cycloid in which the point tracing the curve may be inside the rolling circle (curtate) or outside (prolate). Hypocycloid: variant of a cycloid in which a circle rolls on the inside of another circle instead of a line. Epicycloid: variant of a cycloid in which a circle rolls on the outside of another circle instead of a line. Hypotrochoid: generalization of a hypocycloid where the generating point may not be on the edge of the rolling circle. Epitrochoid: generalization of an epicycloid where the generating point may not be on the edge of the rolling circle. All these curves are roulettes with a circle rolled along another curve of uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi and negative for hypo , then the similitude ratio of curve to evolute is 1 + 2q. The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas. It was also used by Wallace K. Harrison in the design of the Hopkins Center at Dartmouth College in Hanover, New Hampshire. Later work indicates that curtate cycloids do not serve as general models for these curves, which vary considerably.