Epicycloid
Draw epicycloids: the curves a point on the rim of a circle traces as it rolls round the outside of a fixed circle. Choose the fixed and rolling circle as whole numbers, see the cusps, the number of turns before the curve closes, its length and area, and draw the nephroid, the ranunculoid or any ratio from one of 17 presets. Whole-number ratios can also be drawn from straight times-table chords. Watch the circle roll, then download a PDF at real size, a 300 DPI PNG or an SVG for a pen plotter. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). The SVG holds the drawing at the size you typed: the curve as one line, each chord as its own line, for a pen plotter, laser or cutter.
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Runs in your browser with our own code: nothing you type is uploaded or saved. Every line is worked out from the epicycloid formula by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw and print an epicycloid
- Pick a preset or type the fixed circle R and the rolling circle r (whole numbers 1 to 20).
- Tick what to draw: the curve, the fixed and rolling circle, the cusps, or times-table chords (whole-number ratios).
- Press Animate the rolling to watch the circle roll round.
- Pick the colours, line widths, size and paper.
- Download the PDF and print at 100% (Actual size), or the SVG for a plotter or cutter.
What you can use it for
- Maths lessons: roulettes, cusps and ratios
- Gear and cam lessons: rolling-circle curves
- Pen plotter and laser art
- Posters, cards and wall art
- Times-table circle patterns beyond ×2
- Logos and badges built from exact curves
The 17 presets
| Preset | R : r | Cusps | Turns to close | Length | Area |
|---|---|---|---|---|---|
| Cardioid (1:1, 1 cusp) | 1:1 | 1 | 1 | 16r | 6πr² |
| Nephroid (2:1, 2 cusps) | 2:1 | 2 | 1 | 24r | 12πr² |
| 3:1, 3 cusps | 3:1 | 3 | 1 | 32r | 20πr² |
| 4:1, 4 cusps | 4:1 | 4 | 1 | 40r | 30πr² |
| Ranunculoid (5:1, 5 cusps) | 5:1 | 5 | 1 | 48r | 42πr² |
| 6:1, 6 cusps | 6:1 | 6 | 1 | 56r | 56πr² |
| 8:1, 8 cusps | 8:1 | 8 | 1 | 72r | 90πr² |
| 12:1, 12 cusps | 12:1 | 12 | 1 | 104r | 182πr² |
| 3:1 with the rolling circle | 3:1 | 3 | 1 | 32r | 20πr² |
| 3:2 (3 cusps, 2 turns) | 3:2 | 3 | 2 | 40r | – |
| 5:2 (5 cusps, 2 turns) | 5:2 | 5 | 2 | 56r | – |
| 5:3 (5 cusps, 3 turns) | 5:3 | 5 | 3 | 64r | – |
| 7:3 (7 cusps, 3 turns) | 7:3 | 7 | 3 | 80r | – |
| 7:4 (7 cusps, 4 turns) | 7:4 | 7 | 4 | 88r | – |
| 1:2 (1 cusp, 2 turns) | 1:2 | 1 | 2 | 24r | – |
| Nephroid from chords: times table ×3, 200 points | 2:1 | 2 | 1 | 24r | 12πr² |
| 5 cusps from chords: times table ×6, 300 points | 5:1 | 5 | 1 | 48r | 42πr² |
Length and area are in units of the rolling circle radius r. The area is only given for whole-number ratios, where the curve does not cross itself.
Cusps, length and area
| k = R / r | Ratio | Times table | Name | Length 8(R + r) | Area π(R + r)(R + 2r) |
|---|---|---|---|---|---|
| 1 | 1 : 1 | 2 | Cardioid | 16r | 6πr² |
| 2 | 2 : 1 | 3 | Nephroid | 24r | 12πr² |
| 3 | 3 : 1 | 4 | – | 32r | 20πr² |
| 4 | 4 : 1 | 5 | – | 40r | 30πr² |
| 5 | 5 : 1 | 6 | Ranunculoid | 48r | 42πr² |
| 6 | 6 : 1 | 7 | – | 56r | 56πr² |
All values come from the same formulas the drawing uses and are checked in our tests by numerical integration.
The formula
| Part | Meaning |
|---|---|
| (R + r) e^(it) | The centre of the rolling circle, going round at distance R + r |
| − r e^(i(R + r)t/r) | The pen on the rim, turning as the circle rolls without slipping |
| Cusps | Where the pen touches the fixed circle: at t = 2π j q / p |
| Outer circle | The curve stays inside the circle of radius R + 2r and touches it between cusps |
What prints
| Download | What is in it |
|---|---|
| The page at real size, vector: the drawing, R, r, the length and area for your size, and a check bar | |
| PNG 300 DPI | The same page as a picture |
| SVG (real size) | Only the drawing: the curve as one path, each chord and circle as its own element |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| Rolling-circle curves with sharp cusps | Epicycloid (this page) |
| The cardioid three ways, with a worksheet | Cardioid |
| Gear curves with the pen inside the wheel | Spirograph Generator |
| Two sine waves at right angles | Lissajous Curve |
| Banknote-style line patterns | Guilloche Pattern Generator |
Different curves: the Spirograph Generator draws gear curves with the pen inside the wheel, the Guilloche Pattern Generator draws interlaced banknote patterns, and the Fractal Generator draws self-similar shapes.
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Frequently asked questions
What is an epicycloid?
The curve traced by a point on the rim of a circle of radius r rolling round the outside of a fixed circle of radius R. Its formula is x = (R + r) cos t − r cos((R + r) t / r), y = (R + r) sin t − r sin((R + r) t / r).
How many cusps does it have?
When R is a whole number k of times r, there are k cusps and the curve closes after one trip round. When R : r reduces to p : q, there are p cusps and the curve closes after the rolling circle goes round q times. The tool shows both, and our tests check them for every ratio up to 12 : 12.
What are the cardioid, nephroid and ranunculoid?
Epicycloids with 1, 2 and 5 cusps: R = r, R = 2r and R = 5r. The nephroid (kidney shape) was named by Richard A. Proctor in 1878 and the ranunculoid after the buttercup genus Ranunculus. The presets only carry these names when the ratio matches.
How long is an epicycloid?
Each arch, from one cusp to the next, is 8r(R + r)/R long, so the closed curve is 8q(R + r), where q is the number of turns (1 for whole-number ratios). The nephroid is 24r long. Our tests check this against a numerical integration.
What area does it enclose?
For a whole-number ratio, π(R + r)(R + 2r): 6πr² for the cardioid and 12πr² for the nephroid. Curves with more than one turn cross themselves, so the tool gives no single area for them.
How do chords make an epicycloid?
Put N points round a circle and join each point n to point (k + 1)n, taking away N as often as needed. The chords outline an epicycloid with k cusps: ×2 gives a cardioid, ×3 a nephroid. The tool checks that every chord touches the curve.
How is this different from a spirograph?
In a spirograph the pen sits in a hole inside the wheel, so the curve has rounded loops. Here the pen is on the rim of the rolling circle, so the curve has sharp cusps where the pen touches the fixed circle.
What happens with a ratio that is not a fraction?
The curve would never close. This tool uses whole numbers for both circles, so every curve it draws is closed and finite.
Can I use it on a pen plotter, laser or cutter?
Yes. The SVG download is the drawing at the real size you typed: the curve is one unbroken path and every chord or circle is its own element, so a plotter draws each in one stroke. For a program that wants DXF, convert the SVG with our SVG to DXF tool.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every line is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.