Cycloid Calculator

What is a Cycloid?

A cycloid is the curve traced out by a fixed point on the circumference of a circle as the circle rolls along a straight line without slipping. Renowned as the "Helen of Geometers" for its elegant mathematical properties and historical disputes, the cycloid possesses remarkable geometric, kinematic, and physical characteristics.

Parametric Equations of a Cycloid

If a circle of radius $r$ rolls along the horizontal $x$-axis, with the tracing point starting at the origin $(0, 0)$, its coordinates as a function of the rolling parameter $t$ (the angle in radians through which the circle has rotated) are:

$$x(t) = r(t - \sin(t))$$ $$y(t) = r(1 - \cos(t))$$

For one complete revolution of the generating wheel ($0 \le t \le 2\pi$), the curve traces out a single full cycloid arch.

Key Geometric Properties of a Cycloid Arch

For a generating circle of radius $r$:

Base Length (Wavelength): The distance along the base line for one full rotation equals the circumference of the generating circle: $$L = 2\pi r$$
Peak Height (Amplitude): The maximum height occurs at $t = \pi$ (the top of the wheel): $$H_{\text{max}} = 2r$$
Total Arc Length: Remarkably, the total length along the curve of one full arch is an exact integer multiple of the radius: $$S = 8r$$
Area Under the Arch: In 1634, Gilles de Roberval proved that the area under one full cycloid arch is exactly three times the area of the generating circle: $$A = 3\pi r^2 = 3 \times A_{\text{circle}}$$
Radius of Curvature: At any parameter angle $t$, the radius of curvature is given by: $$\rho(t) = 4r \sin\left(\frac{t}{2}\right)$$

Famous Physical Properties

1. The Brachistochrone Problem

In 1696, Johann Bernoulli posed the Brachistochrone problem: what curve allows a bead sliding under gravity without friction to travel between two points in the shortest possible time? The solution is an inverted cycloid.

2. The Tautochrone (Isochrone) Property

Christiaan Huygens discovered that a frictionless particle placed anywhere on an inverted cycloid curve will reach the bottom in the exact same amount of time, regardless of its starting height. Huygens used this property to design the cycloidal pendulum clock.

Frequently Asked Questions

Why is the area under a cycloid arch always 3 times the circle's area?

Integrating the parametric equations $\int y \, dx = \int_0^{2\pi} r(1-\cos t) \cdot r(1-\cos t) \, dt = r^2 \int_0^{2\pi} (1 - 2\cos t + \cos^2 t) \, dt = 3\pi r^2$, which is exactly three times the circle area $\pi r^2$.

What is the difference between a cycloid, curtate cycloid, and prolate cycloid?

A standard cycloid traces a point on the rim ($d = r$). A curtate cycloid (trochoid) traces an interior point ($d < r$, like a point on a wheel spoke), while a prolate cycloid traces an exterior point ($d > r$, like the flange of a train wheel).

How do you calculate the arc length of a cycloid?

For a full single arch, the total arc length is simply $8r$, where $r$ is the radius of the generating circle. For a partial rotation from $0$ to $t$ radians, the length is $s(t) = 4r(1 - \cos(t/2))$.

What tools relate to cycloid curves?

Explore our Circle Sector Calculator, Chord Length Calculator, and Tangential Velocity Calculator.

Report