🌀 Hypocycloids
Rolling circle roulette curves
R : r = 5 : 3
Cusps: 5
Mode
Presets
Geometry
Animation
Color
Controls
Stats
Cusps
5
Progress
0%
Info & Theory

A roulette is the curve traced by a point attached to a circle that rolls without slipping along another curve. Here the rolling circle (radius r) rolls along a fixed circle (radius R).

Hypocycloid (rolling inside)

x = (R−r)cosθ + d·cos((R−r)/r·θ)
y = (R−r)sinθ − d·sin((R−r)/r·θ)

Epicycloid (rolling outside)

x = (R+r)cosθ − d·cos((R+r)/r·θ)
y = (R+r)sinθ − d·sin((R+r)/r·θ)

The pen offset d equals r for the true cusped curve. Smaller d gives a curtate roulette (loops smoothed); larger gives a prolate roulette (extra loops).

Counting cusps

The number of cusps is R / gcd(R, r). The curve closes after r / gcd(R, r) trips around the fixed circle. When R and r share no common factor the cusp count is simply R.

Named special cases

  • R/r = 2 — straight line (Tusi couple / Cardano circles): the point oscillates on a diameter.
  • R/r = 3 — deltoid (3 cusps).
  • R/r = 4 — astroid (4 cusps), x²ᐟ³+y²ᐟ³=R²ᐟ³.
  • Epicycloid r = R — cardioid (1 cusp).
  • Epicycloid R/r = 2 — nephroid (2 cusps).

About Hypocycloids & Epicycloids

A hypocycloid is the curve traced by a point on a small circle rolling inside a larger circle; an epicycloid arises when the rolling circle moves outside. Both are described by the parametric equations x(t) = (R±r)cos(t) ∓ d·cos((R±r)t/r), y(t) = (R±r)sin(t) − d·sin((R±r)t/r), where R is the fixed circle radius, r the rolling circle radius, and d the pen offset from the rolling circle's centre. When d = r the special case curves emerge: the deltoid (R/r = 3), astroid (R/r = 4), cardioid (epicycloid, r = R), and nephroid (r = R/2). These curves appear in optics as caustics — the bright pattern at the bottom of a coffee cup lit from the side is a cardioid.

The simulation animates the rolling gear and traces the path in real time. You can adjust the ratio R/r and the pen offset d to discover named curves and continuous families between them, and observe how the number of cusps equals |R/r| when d = r.

Frequently Asked Questions

What is the difference between a hypocycloid and an epicycloid?

In a hypocycloid the rolling circle moves inside the fixed circle, producing inward-pointing cusps; in an epicycloid it rolls outside, creating outward-pointing loops or petals. The parametric equations differ only in the sign of the rolling term: (R−r) versus (R+r) for the arm length, and the cosine offset sign for x(t).

How many cusps does an astroid have, and what is its area?

The astroid (R/r = 4) has exactly four cusps. Its parametric form is x = R cos³t, y = R sin³t, and its area equals 3πR²/8 — three-eighths of the area of the enclosing circle. The arc length of a full astroid is 6R, which is neatly expressible without trigonometry.

Where do epicycloids appear in the real world?

The cardioid caustic appears in the bottom of a coffee cup when illuminated by a point source. Epicycloidal tooth profiles are used in clock gears because they produce constant angular velocity transmission and minimal friction. The Ptolemaic model of planetary motion used epicycles — small circles rolling on larger ones — to approximate the elliptical orbits observed from Earth.

What is the Tusi couple?

The Tusi couple is the special case where the rolling circle has exactly half the radius of the fixed circle (r = R/2) in a hypocycloid. Every point on the circumference of the rolling circle traces a straight line through the centre of the fixed circle. Named after the 13th-century Persian astronomer Nasir al-Din al-Tusi, it was the first use of a rectilinear motion mechanism derived from circular motion.

What happens when the ratio R/r is irrational?

When R/r is an irrational number the curve never closes — the point traces an endless non-repeating path that densely fills an annular region between two concentric circles. Only when R/r is a rational number p/q (in lowest terms) does the curve close after q revolutions of the rolling circle, forming a curve with p cusps or petals.

What is a hypotrochoid and how does it differ from a hypocycloid?

A hypotrochoid is the general form where the tracing point is at distance d from the centre of the rolling circle, not necessarily on its rim. When d = r you get the hypocycloid (cusps on the fixed circle). When d < r the curve is a rounded star with no cusps; when d > r the loops cross over, producing a more complex self-intersecting shape. The Spirograph toy draws hypotrochoids and epitrochoids using plastic gear rings.

How is a cardioid related to the Mandelbrot set?

The main bulb of the Mandelbrot set is exactly a cardioid. The boundary of the main cardioid is parameterised by c = μ/2 − μ²/4 where μ = e^(2πiθ), which matches the epicycloid with R = r. Points inside this cardioid correspond to values of c for which the iteration z → z² + c converges to a fixed point.

What is the relationship between epicycloids and Fourier series?

Any closed periodic curve can be decomposed into a sum of rotating vectors — epicycles — which is precisely the Fourier series in polar (phasor) form. A single epicycle with frequency ratio R/r produces a simple epicycloid; adding more circles with different radii and frequencies builds up an arbitrary periodic shape. This connection is visualised directly in the Fourier Epicycles simulator.

Can epicycloidal gears transmit constant velocity?

Yes — when both the driving and driven gear tooth profiles are shaped as epicycloids with the pitch circle of each gear as the rolling circle's path, the instantaneous velocity ratio remains constant throughout meshing. This is the theoretical basis of epicycloidal gear design, though in practice involute gears are more common because they are easier to manufacture with consistent tolerances.

About this simulation

Written by MySimulator Team · Reviewed by MySimulator Editorial Review

Last updated: 5 July 2026

This simulator draws roulette curves — the path traced by a point on a small circle as it rolls, without slipping, around a fixed circle. Rolling inside the fixed circle gives a hypocycloid; rolling outside gives an epicycloid. Both follow the classic parametric equations shown in the Info & Theory panel, and the number of cusps the curve ends up with is always R divided by the greatest common divisor of R and r — you can watch that count update live as you change the radii.

🔬 What it shows

The rolling circle (radius r) traces its pen point around the fixed circle (radius R) while the app draws both the guide circles and the accumulated trace. Setting the pen offset d equal to r produces the true cusped curve; smaller d rounds the cusps off into a curtate roulette, larger d produces looping, self-crossing prolate roulettes. Named curves such as the deltoid, astroid, cardioid and nephroid appear automatically at specific R:r ratios and are labelled live above the canvas.

🎮 How to use

Switch Hypocycloid/Epicycloid mode, then either drag Fixed radius R, Rolling radius r and Pen offset d yourself or jump straight to a named curve with the six preset buttons (Tusi line, Deltoid, Astroid, 5-star, Cardioid, Nephroid). Speed controls the drawing rate, the colour buttons switch between violet, rainbow, cyan and gold traces, Trace can be toggled off to watch only the rolling gear, and Save PNG exports the current drawing.

💡 Did you know?

When R is exactly twice r, the hypocycloid collapses to a straight line — the Tusi couple, described by the 13th-century Persian astronomer Nasir al-Din al-Tusi centuries before it was rediscovered in Europe as the Cardano circles. The main bulb of the Mandelbrot set is, remarkably, an exact cardioid — the same curve this simulator draws when the rolling and fixed circles are equal in an epicycloid.

Frequently asked questions

What exactly changes when I switch between Hypocycloid and Epicycloid mode?

The mode only changes the sign in the rolling term of the parametric equations shown in the Info & Theory panel: hypocycloid uses (R minus r) because the small circle rolls inside the fixed one, while epicycloid uses (R plus r) because it rolls outside. Everything else, including the radii, pen offset and cusp-counting rule, works identically in both modes.

How do the Fixed radius R and Rolling radius r sliders decide which named curve appears?

The ratio R divided by r (in its lowest terms) determines the shape entirely. A hypocycloid ratio of 2:1 gives the Tusi line, 3:1 gives the deltoid, and 4:1 gives the astroid; an epicycloid ratio of 1:1 gives the cardioid and 2:1 gives the nephroid. Any other ratio still produces a valid roulette, just without a classical name, and the live label above the canvas announces a match whenever one is found.

What does the Pen offset d slider actually do?

It sets how far the tracing point sits from the centre of the rolling circle, relative to its radius r. At d equal to 1.00 (that is, d = r) you get the true cusped hypocycloid or epicycloid; below that the loops smooth out into a rounded curtate roulette with no sharp points, and above it the tracing point swings wide enough to create self-intersecting prolate loops.

How is the live Cusps number calculated?

The simulator computes the greatest common divisor of the Fixed radius R and Rolling radius r, then divides R by that value, which gives the exact number of cusps the curve will show once it closes. The same divisor also tells you how many full trips the rolling circle makes around the fixed circle before the pattern repeats.

What is special about the Tusi line preset?

With R exactly twice r, every point on the rolling circle's rim traces a straight line through the centre of the fixed circle instead of a curve with cusps. This mechanism, now called the Tusi couple, was devised by the astronomer Nasir al-Din al-Tusi to generate straight-line motion from purely circular motion, without needing Ptolemy's disputed equant point.