The Cycloid: The Curve Traced by a Rolling Wheel

The construction: a point on a rolling wheel

Take a circle of radius r sitting on the x-axis and mark one point P on its rim, starting at the origin. Now roll the circle to the right without slipping. “Without slipping” is the whole secret: after the wheel turns through an angle t (in radians), the length of rim that has unrolled onto the ground is exactly the arc length rt, so the center has advanced to (rt, r). The marked point P is at the center plus a vector of length r that has rotated clockwise by t, giving the classic parametric equations:

• x(t) = r(t − sin t)
• y(t) = r(1 − cos t)

At t = 0 the point is on the ground at the origin. At t = π the wheel has done a half turn and P is at the top of its arch, (πr, 2r). At t = 2π it returns to the ground at (2πr, 0), and a fresh arch begins. The mixture of the linear term t and the trigonometric term sin t means the cycloid is a transcendental curve — there is no polynomial equation in x and y that captures it.

Why the rim point stops: the cusp

Differentiate the parametrization to get the velocity of the tracing point: x′ = r(1 − cos t) and y′ = r sin t. Its speed is

|P′| = r√[(1 − cos t)² + sin²t] = r√(2 − 2 cos t) = 2r |sin(t/2)|.

At t = 0, 2π, 4π… this speed is zero. The point that is touching the ground is momentarily at rest — the exact same fact that lets a rolling tire grip the road. The contact point is the wheel’s instantaneous center of rotation, so every rim point passing through it has vanishing velocity, and the curve there forms a sharp cusp (a corner with a vertical tangent), not a smooth bottom. At the top of the arch, t = π, the speed reaches its maximum 2r — twice the speed of the center. That is why the top of a moving wheel is a blur while the bottom looks frozen in a photograph. The arch itself is 2πr wide and 2r tall: its height is the wheel’s diameter, not its circumference.

Arc length is exactly 8r — with no π in sight

Christopher Wren proved in 1658 that the length of one full arch is exactly eight radii. We can see it directly by integrating the speed:

L = ∫₀^2π 2r sin(t/2) dt = 2r [−2 cos(t/2)]₀^2π = 2r (2 + 2) = 8r.

(On the interval [0, 2π], sin(t/2) is non-negative, so the absolute value drops away.) The astonishing feature is that no π survives, even though the curve was born from a rolling circle. This was one of the very first rectifications — exact lengths of a curved line — in the history of mathematics, achieved before calculus was formalized. It says a wheel of radius 1 metre lays down an arch of curve exactly 8 metres long every time it turns once, while advancing only 2π ≈ 6.28 metres along the road.

The area is exactly three rolling circles

Galileo, around 1599, guessed the arch’s area by cutting cycloids and circles out of sheet metal and weighing them; he found the ratio was close to 3 but suspected it was irrational. It is not — it is exactly 3. Integrating gives the clean answer at once:

A = ∫ y dx = ∫₀^2π r(1 − cos t) · r(1 − cos t) dt = r² ∫₀^2π (1 − cos t)² dt = 3πr².

Since the generating circle has area πr², the arch encloses exactly three times as much. Gilles de Roberval proved this rigorously in 1634 without calculus, using his “companion curve” — the sine curve x = rt, y = r(1 − cos t). By Cavalieri’s principle, at every height the horizontal gap between the cycloid and its companion equals a semichord (half-chord) of the rolling circle — the horizontal distance from its center to the rim — so the region between the two curves has the area of the whole circle, πr². The companion curve itself neatly bisects the circumscribing 2πr × 2r rectangle, contributing 2πr². Adding them: 2πr² + πr² = 3πr².

Tautochrone and brachistochrone: two prizes on one curve

Flip the cycloid upside down (cusps pointing up, a valley in the middle) and it acquires two almost magical dynamical properties.

Tautochrone (equal-time curve). Christiaan Huygens showed in 1659 that a frictionless bead released from anywhere on the inverted cycloid reaches the bottom in the same time. Measuring arc length s from the lowest point, the height turns out to be proportional to s², which makes the along-curve motion obey s″ = −(g/4r) s — pure simple harmonic motion, whose period ignores amplitude. The descent time is π√(r/g) no matter where you start. Huygens exploited this to design a genuinely isochronous pendulum clock: he hung the bob between two cycloidal “cheeks,” because the curve a swinging string wraps around them (its involute) is itself a cycloid.

Brachistochrone (fastest-descent curve). In 1696 Johann Bernoulli challenged Europe to find the shape of wire down which a bead slides between two points in the least time. The answer is again the inverted cycloid — a longer, plunging path that beats the straight line by building up speed early. Bernoulli’s elegant solution treated the bead like a ray of light obeying Fermat’s least-time principle, so that sin θ / v stays constant (Snell’s law); with v = √(2gy) this differential condition integrates to the cycloid. Newton reportedly solved it overnight and submitted anonymously; Bernoulli recognized the author instantly, “as one knows the lion by his claw.” The problem seeded the entire field of the calculus of variations.

Relatives: the trochoid family and the evolute

The cycloid is the pure case of a broader family. If the tracing point lies inside the rim (distance d < r from center), you get a curtate cycloid — a gentle wave with no cusps, the path of a point on the hub of a rolling wheel. If it lies outside the rim (d > r, like the flange of a train wheel dipping below the rail), you get a prolate cycloid, which loops backward at the bottom. General equations: x = rt − d sin t, y = r − d cos t. Roll the circle on another circle instead of a line and the rim point sweeps an epicycloid (outside — the heart-shaped cardioid, the two-lobed nephroid) or a hypocycloid (inside — the three-cusped deltoid, the four-cusped astroid).

A deeper self-reference makes the pendulum trick work: the evolute of a cycloid — the curve traced by its centers of curvature — is an identical cycloid, merely shifted by half a period and dropped by 2r. Its radius of curvature is 4r sin(t/2), reaching 4r at the top of the arch. Because a cycloid’s evolute is a cycloid, its involute is too, which is exactly the geometric fact Huygens needed to force his pendulum bob onto a tautochrone path.

History: the Helen of geometers

For a century the cycloid was the battlefield of the finest mathematicians in Europe, and the priority disputes and accusations of plagiarism it provoked earned it the nickname the Helen of geometers — beautiful, and the cause of endless quarrels. Galileo named it and estimated its area; Marin Mersenne circulated the problems; Roberval proved the area (1634) but kept his methods secret, later fueling a plagiarism row with Torricelli. Descartes and Fermat found the tangent independently. In 1658 Blaise Pascal, sleepless with a toothache, solved a battery of cycloid problems — areas, centers of gravity, volumes of revolution — and issued them as a public prize under the pseudonym Amos Dettonville; the same year Wren rectified the arch at 8r. Huygens crowned the classical era with the tautochrone and his 1673 Horologium Oscillatorium, and the Bernoullis and Newton closed it with the brachistochrone. Along the way the cycloid drove the invention of the calculus of variations and seeded modern ideas about optimal paths, from the geodesics of general relativity to the shape of the fastest ski-jump ramp.

The cycloid sits inside the trochoid family — curves drawn by a point at distance d from the center of a rolling circle of radius r, on a line or on another circle.

Curve	Generating point	Distinctive feature	Cusps / loops
Cycloid	on the rim (d = r), circle on a line	sharp cusps, arch 8r long	cusp each turn
Curtate cycloid	inside the rim (d < r)	smooth low waves, never touches line	no cusp, no loop
Prolate cycloid	outside the rim (d > r)	point loops backward at the bottom	a loop each turn
Epicycloid	on rim, circle rolls outside a circle	cardioid (1 cusp), nephroid (2)	cusps point outward
Hypocycloid	on rim, circle rolls inside a circle	deltoid (3), astroid (4 cusps)	cusps point inward