The cycloid: the curve of a rolling wheel

Mark a point on the rim of a wheel and roll it along a straight line: the point traces a cycloid, a row of arches with sharp cusps where the point touches the ground. It is also the answer to one of the most famous problems in mathematics: the shape of the fastest slide.

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Three arches of the cycloid (t − sin t, 1 − cos t), traced by a wheel of radius 1.

Equations to type

Cycloid (radius r = 1)

(t - sin(t), 1 - cos(t))

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Any radius a, with a slider

(a(t - sin(t)), a(1 - cos(t))) a = 1.5

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Prolate (point outside the rim)

(t - 2sin(t), 1 - 2cos(t))

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At a glance

Parametric equations: x = r(t − sin t), y = r(1 − cos t)
Length of one arch: 8r
Area under one arch: 3πr², three times the wheel
Width of one arch: 2πr

Where the equations come from

After the wheel has turned through angle t, its centre has moved rt to the right and sits at (rt, r). The marked point has rotated by t from the bottom, so it is at an offset (−r sin t, −r cos t) from the centre. Adding the two gives x = r(t − sin t), y = r(1 − cos t).

The fastest slide

In 1696 Johann Bernoulli challenged mathematicians to find the curve along which a bead slides between two points in the shortest time, under gravity alone. The straight line is shortest but not fastest; the answer is an upside-down cycloid, which drops steeply at first to pick up speed. Newton, Leibniz, l’Hôpital and Jakob Bernoulli all solved it, and the problem helped found the calculus of variations.

The cycloid has a second magic property: a bead released anywhere on an inverted cycloid reaches the bottom in the same time. Christiaan Huygens used this tautochrone property to design more accurate pendulum clocks.

Area and length

One arch has length exactly 8r, four times the diameter of the wheel, a result found by Christopher Wren in 1658. The area under one arch is 3πr², three times the area of the wheel, which Gilles de Roberval proved around 1634. In the calculator, int((1 - cos(t))^2, t, 0, 2pi) gives the area under one arch for r = 1, since y dx = (1 − cos t)² dt, and it returns 9.42477796077, which is 3π.