Cycloid Calculator - Parametric Curve Properties

A cycloid is a remarkable curve traced by a point fixed on the rim of a circle as the circle rolls along a straight line without slipping. It was named and first studied seriously by Galileo Galilei in the early seventeenth century, and it later captured the attention of Blaise Pascal, the Bernoulli brothers, Christiaan Huygens, and Isaac Newton. Despite its simple mechanical origin, the cycloid possesses a surprising set of geometric and physical properties that make it one of the most significant curves in the history of mathematics. The parametric equations that define the cycloid are x = r(t − sin t) and y = r(1 − cos t), where r is the radius of the rolling circle and t is the angle, measured in radians, through which the circle has rotated. When t = 0 the traced point is at the origin, touching the line on which the circle rolls. As t increases from 0 to 2π the point sweeps out one complete arch, rising to its peak height of 2r when t = π, and returning to the baseline at x = 2πr when t = 2π. This cycle repeats indefinitely as the circle continues rolling, producing a series of identical arches. One of the most striking properties of the cycloid is the length of a single arch. Whereas the circumference of the generating circle is 2πr, the arc length of one cycloid arch is exactly 8r — four times the diameter, or approximately 2.546 times the circumference. This result, first proved by Christopher Wren in 1658, surprised mathematicians of the era because it yielded a clean rational multiple of the radius rather than an irrational multiple involving π. Equally notable is the area under one arch. It equals 3πr², which is precisely three times the area of the generating circle πr². This was established by Gilles de Roberval in 1634 and was one of the first significant results obtained by methods that anticipated integral calculus. The cycloid is also the solution to two famous variational problems. The brachistochrone problem, posed by Johann Bernoulli in 1696, asks for the curve of fastest descent under gravity between two points not on a vertical line; the answer is a cycloid. The tautochrone problem asks for a curve on which an object slides to the bottom in the same time regardless of its starting position; the answer is again a cycloid. Huygens used the tautochrone property to design cycloidal pendulum clocks that kept more accurate time than ordinary pendulums. In engineering, cycloidal profiles appear in gear teeth, cam mechanisms, and compact gear reducers called cycloidal drives. In robotics, high-reduction cycloidal gearboxes provide precise torque transmission in a small package. Computer graphics and animation use cycloidal and epicycloidal curves to generate organic-looking motion paths. This calculator lets you explore all these properties by entering any positive radius and any parameter value.