Hypocycloid
Draw hypocycloids: the curves a point on the rim of a circle traces as it rolls round the inside 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 deltoid, the straight-line case or any ratio from one of 17 presets. Add the curve’s tangent lines to see it appear from straight lines. 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 ladder or tangent line 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 hypocycloid formula by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw and print a hypocycloid
- Pick a preset or type the fixed circle R and the rolling circle r (whole numbers, r smaller than R).
- Tick what to draw: the curve, the fixed and rolling circle, the cusps, or tangent lines.
- Press Animate the rolling to watch the circle roll round inside.
- 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 mechanism lessons: a circle rolling inside a ring
- Pen plotter and laser art
- Posters, cards and wall art
- Star shapes with any number of points
- Logos and badges built from exact curves
The 17 presets
| Preset | R : r | Cusps | Turns to close | Length | Area |
|---|---|---|---|---|---|
| 2:1, a straight line (2 cusps) | 2:1 | 2 | 1 | 8r | – |
| Deltoid (3:1, 3 cusps) | 3:1 | 3 | 1 | 16r | 2πr² |
| Astroid (4:1, 4 cusps) | 4:1 | 4 | 1 | 24r | 6πr² |
| 5:1, 5 cusps | 5:1 | 5 | 1 | 32r | 12πr² |
| 6:1, 6 cusps | 6:1 | 6 | 1 | 40r | 20πr² |
| 8:1, 8 cusps | 8:1 | 8 | 1 | 56r | 42πr² |
| 12:1, 12 cusps | 12:1 | 12 | 1 | 88r | 110πr² |
| 3:1 with the rolling circle | 3:1 | 3 | 1 | 16r | 2πr² |
| 3:2 (3 cusps, 2 turns) | 3:2 | 3 | 2 | 8r | – |
| 5:2 (5 cusps, 2 turns) | 5:2 | 5 | 2 | 24r | – |
| 7:2 (7 cusps, 2 turns) | 7:2 | 7 | 2 | 40r | – |
| 7:3 (7 cusps, 3 turns) | 7:3 | 7 | 3 | 32r | – |
| 8:3 (8 cusps, 3 turns) | 8:3 | 8 | 3 | 40r | – |
| 9:5 (9 cusps, 5 turns) | 9:5 | 9 | 5 | 32r | – |
| 3:1 from 60 tangent lines | 3:1 | 3 | 1 | 16r | 2πr² |
| 5:1 from 120 tangent lines | 5:1 | 5 | 1 | 32r | 12πr² |
| 5:2 from 150 tangent lines, with the curve | 5:2 | 5 | 2 | 24r | – |
Length and area are in units of the rolling circle radius r. The area is only given for whole-number ratios of 3 or more, where the curve does not cross itself.
Cusps, length and area
| k = R / r | Ratio | Cusps | Name | Length 8(k − 1)r | Area (k − 1)(k − 2)πr² |
|---|---|---|---|---|---|
| 2 | 2 : 1 | 2 | Straight line | 8r | 0 (a segment) |
| 3 | 3 : 1 | 3 | Deltoid | 16r | 2πr² |
| 4 | 4 : 1 | 4 | Astroid | 24r | 6πr² |
| 5 | 5 : 1 | 5 | – | 32r | 12πr² |
| 6 | 6 : 1 | 6 | – | 40r | 20πr² |
| 8 | 8 : 1 | 8 | – | 56r | 42πr² |
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 the other way as the circle rolls without slipping |
| Cusps | Where the pen touches the fixed circle: at t = 2π j q / p |
| Inner circle | The curve stays between the circle of radius |R − 2r| and the fixed circle |
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 tangent line 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, rolling inside | Hypocycloid (this page) |
| The four-cusped curve with the sliding ladder | Astroid |
| Rolling round the outside (cardioid, nephroid) | Epicycloid |
| Gear curves with the pen inside the wheel | Spirograph Generator |
| Two sine waves at right angles | Lissajous Curve |
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 a hypocycloid?
The curve traced by a point on the rim of a circle of radius r rolling round the inside 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 centre of the rolling circle goes round q times. The tool shows both, and our tests check them for every ratio up to 12.
What is a deltoid?
The hypocycloid with three cusps, R = 3r. Its area is 2πr² and its length 16r. The preset only carries the name when the ratio is 3 : 1 (or the same ratio, such as 6 : 2).
What happens when R is twice r?
The pen moves back and forth along a straight line, a diameter of the fixed circle. This pair of circles is called a Tusi couple, and the tool draws it as a straight line, which our tests check.
How long is a hypocycloid?
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): 8r(k − 1) for k cusps. Our tests check this against a numerical integration.
What area does it enclose?
For a whole-number ratio of 3 or more, π(R − r)(R − 2r), which is (k − 1)(k − 2)πr². Curves with more than one turn cross themselves, so the tool gives no single area for them.
Why do two different ratios give the same curve?
A circle of radius r and a circle of radius R − r rolling inside the same fixed circle draw the same curve, going round opposite ways. So 3 : 1 and 3 : 2 both draw the deltoid. The tool says so and our tests check it.
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.
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 ladder or tangent line 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.