Cycloid

A cycloid is the curve traced by a fixed point on the rim of a circle of radius r as the circle rolls without slipping along a straight line. The standard cycloid is obtained when the tracing point lies on the circumference; its parametric equations are x = r(t − sin t) and y = r(1 − cos t), with t representing the angle through which the circle has rotated.

Key properties arise from these equations. Consecutive cusps occur at t = 2πk, so the horizontal distance between cusps is 2πr, and the maximum vertical height of the arch is 2r. The slope is dy/dx = sin t/(1 − cos t) = cot(t/2), which tends to infinity at cusps. One arch of a cycloid (between two cusps) has length 8r, and the area under one arch is 3πr^2.

To describe a broader family rather than a single cycloid, a tracing point at distance d from the center of the rolling circle yields a trochoid with parametric form x = r t − d sin t, y = r − d cos t. When d = r, this reduces to the cycloid. If d < r, the curve is a curtate cycloid (loops do not occur); if d > r, it is a prolate cycloid (loops can occur).

Historically, the cycloid is notable in the study of the brachistochrone and tautochrone problems; the cycloid is the curve of fastest descent under gravity and has the property that the time of descent to the bottom is independent of the starting point. It also appears in engineering contexts, such as cycloidal gears and rollers in certain mechanical drives.