Cycloid

Cycloid

The path traced by a point on a wheel as the wheel rolls, without slipping, along a flat surface. The standard parametrization is x = a(t – sin t), y = a(1 – cos t), where a is the radius of the wheel.

Note: Cycloids are periodic functions.

Cycloid curve on x-y axes showing repeating arch-shaped loops traced by a rolling wheel, with a small circle visible near the...

Key Formula

x = a(t - \sin t), \quad y = a(1 - \cos t)

Where:

$x$ = Horizontal coordinate of the point on the cycloid
$y$ = Vertical coordinate of the point on the cycloid
$a$ = Radius of the rolling circle
$t$ = Parameter representing the angle (in radians) through which the circle has rotated

Worked Example

Problem: A circle of radius 2 rolls along a flat surface. Find the coordinates of the point on the rim when the circle has rotated through t = π/2, t = π, and t = 2π.

Step 1: Write the parametric equations with a = 2.

x = 2(t - \sin t), \quad y = 2(1 - \cos t)

Step 2: At t = π/2: sin(π/2) = 1 and cos(π/2) = 0.

x = 2\!\left(\frac{\pi}{2} - 1\right) = \pi - 2 \approx 1.14, \quad y = 2(1 - 0) = 2

Step 3: At t = π: sin π = 0 and cos π = −1.

x = 2(\pi - 0) = 2\pi \approx 6.28, \quad y = 2(1 - (-1)) = 4

Step 4: At t = 2π: sin 2π = 0 and cos 2π = 1. The point returns to the ground, completing one full arch.

x = 2(2\pi - 0) = 4\pi \approx 12.57, \quad y = 2(1 - 1) = 0

Answer: The coordinates are approximately (1.14, 2) at t = π/2, (6.28, 4) at t = π (the top of the arch), and (12.57, 0) at t = 2π (back on the ground). The arch height equals 2a = 4, and the arch width equals 2πa = 4π.

Another Example

This example applies parametric integration to a cycloid rather than simply evaluating coordinates. It demonstrates the famous result that the area under one arch equals three times the area of the rolling circle.

Problem: Find the area under one arch of a cycloid with radius a = 3.

Step 1: The area under one arch of a parametric curve is given by the integral from t = 0 to t = 2π.

A = \int_0^{2\pi} y\,\frac{dx}{dt}\,dt

Step 2: Compute dx/dt from x = 3(t − sin t).

\frac{dx}{dt} = 3(1 - \cos t)

Step 3: Substitute y = 3(1 − cos t) and dx/dt into the integral.

A = \int_0^{2\pi} 3(1 - \cos t) \cdot 3(1 - \cos t)\,dt = 9\int_0^{2\pi}(1 - \cos t)^2\,dt

Step 4: Expand and integrate. Use the identity cos²t = (1 + cos 2t)/2.

\int_0^{2\pi}(1 - 2\cos t + \cos^2 t)\,dt = \int_0^{2\pi}\!\left(\frac{3}{2} - 2\cos t + \frac{\cos 2t}{2}\right)dt = 3\pi

Step 5: Multiply by the coefficient of 9.

A = 9 \cdot 3\pi = 27\pi

Answer: The area under one arch is 27π ≈ 84.82 square units. In general, the area under one arch of a cycloid is 3πa², which is exactly three times the area of the generating circle.

Frequently Asked Questions

What is special about a cycloid curve?

The cycloid has two remarkable physical properties. It is the brachistochrone — the curve of fastest descent between two points under gravity. It is also the tautochrone — the curve along which a frictionless bead reaches the bottom in the same time regardless of where it starts. These properties made the cycloid a central topic in the history of calculus and physics.

What is the difference between a cycloid and a cardioid?

A cycloid is traced by a point on a circle rolling along a straight line, and it produces a series of arches in the Cartesian plane. A cardioid is traced by a point on a circle rolling around another circle of equal radius, producing a heart-shaped closed curve in polar coordinates. Both involve rolling circles, but the surface they roll on — flat line versus another circle — is different.

How do you find the arc length of one arch of a cycloid?

Using the parametric arc length formula, one arch of a cycloid with radius a has length 8a. You compute ds/dt = √((dx/dt)² + (dy/dt)²) = 2a|sin(t/2)| and integrate from 0 to 2π. This clean result is another remarkable property of the cycloid.

Cycloid vs. Cardioid

	Cycloid	Cardioid
Definition	Point on a circle rolling along a straight line	Point on a circle rolling around a fixed circle of equal radius
Equation type	Parametric: x = a(t − sin t), y = a(1 − cos t)	Polar: r = a(1 + cos θ)
Shape	Repeating arches (open curve)	Heart-shaped closed curve
Area	3πa² per arch	3πa²/2 total enclosed area
Periodicity	Periodic with period 2πa along x-axis	Closed curve, period 2π in θ

Why It Matters

You encounter cycloids in calculus courses when studying parametric equations, arc length, and areas under parametric curves. In physics, the cycloid appears as the solution to both the brachistochrone and tautochrone problems, which were landmark challenges that helped develop the calculus of variations. Understanding the cycloid also builds intuition for how rolling motion generates curves, a concept that extends to epicycloids, hypocycloids, and gear design.

Common Mistakes

Mistake: Confusing the parameter t with the x-coordinate or treating t as time.

Correction: The parameter t is the angle (in radians) through which the circle has rotated. The x-coordinate is a(t − sin t), not t itself. While t can represent time if the circle rotates at one radian per second, in the standard parametrization it is simply a geometric angle.

Mistake: Assuming the maximum height of the cycloid equals the radius a.

Correction: The highest point occurs at t = π, where y = a(1 − cos π) = 2a. The peak of each arch is at height 2a — twice the radius — because the tracing point rises to the very top of the rolling circle, which itself sits one radius above the ground.

Related Terms

Parametrize — Cycloid is defined by parametric equations
Brachistochrone — Cycloid is the curve of fastest descent
Tautochrone — Cycloid is the curve of equal descent time
Cardioid — Related rolling-circle curve on a circle
Periodic Function — Cycloid repeats with period 2πa
Area Using Parametric Equations — Used to find area under a cycloid arch
Trig Functions — Sine and cosine appear in cycloid equations
Radius of a Circle or Sphere — Parameter a is the radius of the rolling circle