About

A cycloid is the curve traced by a point on the rim of a circle rolling along a straight line. Miscalculating its arc length or area leads to errors in gear tooth profiles, cam mechanisms, and brachistochrone problem solutions. This tool computes exact properties for five cycloid families: ordinary, curtate (tracing point inside the circle, d < r), prolate (tracing point outside, d > r), epicycloid (circle rolling outside another circle), and hypocycloid (circle rolling inside). All geometric quantities - arc length, enclosed area, radius of curvature - are derived from parametric integrals, not approximations. Numerical integration uses Simpson's rule with 1000 subdivisions for arc length where closed-form solutions do not apply.

Limitations: the tool assumes perfect rolling contact with no slip. Epicycloid and hypocycloid calculations assume integer or rational radius ratios Rr for closed curves. Irrational ratios produce space-filling curves that never close. The visualizer renders up to 50 full revolutions to approximate such cases.

Formulas

The ordinary cycloid is defined by a circle of radius r rolling on a line. The parametric equations are:

x(t) = r(t − sin t)

y(t) = r(1 − cos t)

For curtate and prolate cycloids, the tracing point is at distance d from center:

x(t) = rt − d sin t

y(t) = r − d cos t

For epicycloids (small circle of radius r rolls outside fixed circle of radius R):

x(t) = (R + r)cos t − r cos((R + r)rt)

y(t) = (R + r)sin t − r sin((R + r)rt)

For hypocycloids (small circle rolls inside):

x(t) = (R − r)cos t + r cos((R − r)rt)

y(t) = (R − r)sin t − r sin((R − r)rt)

Arc length for one complete arch is computed via:

L = 2π∫0 √x′(t)² + y′(t)² dt

For the ordinary cycloid this evaluates to L = 8r. Curvature at parameter t:

κ(t) = |x′y″ − y′x″|(x′² + y′²)^3/2

Where r = rolling circle radius, R = fixed circle radius, d = tracing point distance from center, t = parameter (angle of rotation), k = R/r = radius ratio, κ = curvature, L = arc length.

Reference Data

Cycloid Type	Parameters	Arc Length (1 arch)	Area (1 arch)	Notable Property
Ordinary	r	8r	3πr²	Brachistochrone & tautochrone curve
Curtate	r, d < r	Numerical integral	Numerical integral	No cusps, smooth loops
Prolate	r, d > r	Numerical integral	Numerical integral	Self-intersecting loops
Epicycloid (k=1)	R=r	8r	3πr²	Cardioid
Epicycloid (k=2)	R=2r	16r	8πr²	Nephroid
Epicycloid (k=3)	R=3r	24r	15πr²	Trefoil / 3-cusped epicycloid
Hypocycloid (k=3)	R=3r	24r3	2πr²3	Deltoid (Steiner curve)
Hypocycloid (k=4)	R=4r	6R	3πR²8	Astroid
Hypocycloid (k=5)	R=5r	Numerical integral	Numerical integral	Pentacuspid
Epicycloid (general)	R, r	8r(R + r)R	πr²⋅(3k + k²)	Closed when k = R/r ∈ Q
Hypocycloid (general)	R, r	8r(R − r)R	πr²⋅(3k − k²)	Requires R > r
Involute of circle	r	rt²2	N/A (open curve)	Used in gear tooth profiles
Spirograph (general)	R, r, d	Numerical integral	Numerical integral	Generalized hypotrochoid / epitrochoid

Frequently Asked Questions

In a curtate cycloid the tracing point distance d is less than the rolling radius r, producing a smooth undulating curve without cusps. In a prolate cycloid d > r, causing the curve to form self-intersecting loops. When d = r you get the ordinary cycloid with sharp cusps.

A closed epicycloid requires the rolling circle to return to its starting position, which happens after k = R/r revolutions. When k is irrational, the curve never exactly repeats and densely fills an annular region. The tool approximates this by rendering up to 50 full revolutions of the parameter.

The tool uses composite Simpson's rule with 1000 subintervals, yielding accuracy to approximately 10 significant digits for smooth parametric curves. For ordinary cycloids, the closed-form result L = 8r is used directly. Numerical integration is applied only to curtate, prolate, and generalized cases where no elementary closed form exists.

At cusps (t = 0, 2π, 4π, ...) both derivatives x′ and y′ equal zero, making the curvature formula indeterminate. The limit of the radius of curvature approaches zero at these points. The tool evaluates curvature at t = π (top of the arch) where it equals 14r.

Galileo conjectured and Roberval proved that the area under one arch of an ordinary cycloid is exactly 3πr², which is three times the area of the generating circle. This elegant ratio holds regardless of the circle size and was historically significant in the development of integral calculus.

Yes. A Spirograph produces hypotrochoids (point inside a circle rolling inside a larger circle at distance d from center) and epitrochoids (rolling outside). Select the Epicycloid or Hypocycloid type and adjust the tracing distance d to differ from the rolling radius r. The resulting curve is a generalized trochoid identical to Spirograph output.