Trochoid
Draw trochoids: the curves a point at distance d from the centre of a circle traces as the circle rolls along a straight line, x = rt − d sin t, y = r − d cos t. With the pen inside the circle (d < r) you get a smooth wave, on the rim (d = r) the cycloid with its cusps, and outside (d > r) a curve with one loop per turn. Show the rolling circle and pen arm, the circle at several places, where each loop crosses itself, and several pens at once, or turn the curve upside down for a trochoidal wave profile. Start from one of 14 presets, then download a PDF at real size, a 300 DPI PNG or an SVG. 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, each curve as one line, for a pen plotter, laser or cutter.
Trochoid
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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 trochoid formula by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw and print a trochoid
- Pick a preset or type the pen distance d as a multiple of the circle radius r (0.05 to 4).
- Choose how many periods (turns of the circle) to draw.
- Tick the rolling circle and pen arm, the circle at several places or the loop crossings, and type more d values to draw several pens at once.
- Press Animate the rolling to watch the circle roll.
- Pick the colours, size and paper, then 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: curtate, cycloid and prolate side by side
- Physics and engineering lessons: a point on a rolling wheel
- Pen plotter, laser and vinyl art: loops and waves
- Borders, ribbons and repeat strips for print
- Wave profiles for posters and diagrams
- Calculus classes: arc length by numerical integration
The 14 presets
| Preset | Pen distance d | Kind | Periods | Upside down |
|---|---|---|---|---|
| Curtate, d = 0.5r, with the rolling circle | 0.5r | curtate | 2 | no |
| Curtate, d = 0.25r, three periods | 0.25r | curtate | 3 | no |
| Curtate, d = 0.75r | 0.75r | curtate | 2 | no |
| d = r: the cycloid (cusps) | 1r | cycloid | 2 | no |
| Prolate, d = 1.5r, with the rolling circle | 1.5r | prolate | 2 | no |
| Prolate, d = 2r, crossings marked | 2r | prolate | 2 | no |
| Prolate, d = 3r, big loops | 3r | prolate | 2 | no |
| Curtate, cycloid and prolate together | 0.5r, 1r, 1.5r | curtate, cycloid, prolate | 2 | no |
| Five pens from d = 0.25r to 2r | 0.25r, 0.5r, 1r, 1.5r, 2r | curtate, cycloid, prolate | 2 | no |
| Prolate, d = 1.5r, the circle at 9 places | 1.5r | prolate | 1 | no |
| Curtate, d = 0.5r, the circle at 9 places | 0.5r | curtate | 1 | no |
| Trochoidal wave profile (upside down), d = 0.5r | 0.5r | curtate | 4 | yes |
| Trochoidal wave profile (upside down), d = 0.9r | 0.9r | curtate | 4 | yes |
| Prolate, d = 1.25r, six periods | 1.25r | prolate | 6 | no |
Each preset starts with the pen at the top of the circle (t = −π), so every period and every loop is drawn whole.
Curtate, cycloid and prolate
| d | Kind | Loops | Length of one period | Area to the line |
|---|---|---|---|---|
| 0.25r | curtate | none | 6.382r | 2.0625πr² |
| 0.5r | curtate | none | 6.682r | 2.25πr² |
| 0.75r | curtate | none | 7.204r | 2.5625πr² |
| 1r | cycloid | cusps on the line | 8.000r | 3πr² |
| 1.25r | prolate | one per period | 9.172r | – |
| 1.5r | prolate | one per period | 10.505r | – |
| 2r | prolate | one per period | 13.365r | – |
| 3r | prolate | one per period | 19.377r | – |
Lengths come from numerical integration, areas from π(2r² + d²); both are checked in our tests. Multiply by your r in mm to get real sizes; the PDF prints them for you.
The formula
| Part | Meaning |
|---|---|
| (rt, r) | The centre of the rolling circle after it turns through t: it has rolled a distance rt |
| − d (sin t, cos t) | The pen, at distance d from the centre, turning with the circle |
| d < r | Curtate: x always increases, no loops |
| d = r | The cycloid: cusps on the line every 2πr |
| d > r | Prolate: one self-crossing per period |
What prints
| Download | What is in it |
|---|---|
| The page at real size, vector: the drawing, r, d, 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: each pen’s curve as one path, circles, arms, dots |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| A circle rolling on a line, pen anywhere | Trochoid (this page) |
| The pen exactly on the rim, with the brachistochrone | Cycloid |
| Rolling round a circle, pen inside the wheel (hypotrochoids, epitrochoids) | Spirograph Generator |
| Two sine waves at right angles | Lissajous Curve |
| Smooth web section dividers | SVG Wave Generator |
Different tools: the Spirograph Generator rolls gear wheels round a ring, the SVG Wave Generator makes layered waves for web pages, and the Guilloche Pattern Generator draws interlaced banknote patterns.
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Frequently asked questions
What is a trochoid?
The curve traced by a point fixed to a circle as the circle rolls along a straight line without slipping. If the circle has radius r and the point is at distance d from its centre, then after the circle turns through t the point is at x = rt − d sin t, y = r − d cos t.
What are curtate and prolate cycloids?
With the pen inside the circle (d < r) the curve is a curtate trochoid, or curtate cycloid: a smooth wave that never loops. With the pen outside the circle (d > r) it is a prolate trochoid, or prolate cycloid, with one loop for every turn of the circle. With d = r it is the ordinary cycloid. Some books swap the two names, so the tool always shows d as well.
Why does a prolate trochoid make loops?
When the pen is further from the centre than the rim, it moves backwards for part of each turn: at the bottom of the circle it travels against the direction of rolling. So the curve doubles back and crosses itself once per turn. Our tests count the crossings for each d and each number of periods.
How long is one period of a trochoid?
There is no simple formula: the length is the integral of √(r² + d² − 2rd cos t) over one turn, which the tool works out numerically. Our tests check it against a second integration and an elliptic-integral formula. For d = r it is 8r, the cycloid arch.
What area does a trochoid enclose?
For d up to r, the area between one period and the line the circle rolls on is π(2r² + d²): 3πr² for the cycloid. Prolate curves loop below the line, so the tool gives no single area for them.
What is the trochoidal wave?
The trochoidal (or Gerstner) wave is a classical model of water waves whose surface is an upside-down trochoid with the pen inside the circle: sharp crests and flat troughs. Tick Upside down with d below r to draw that profile. With d = r the crests become cusps.
Does it draw hypotrochoids and epitrochoids?
No. Those come from a circle rolling inside or outside another circle, which is what our Spirograph Generator draws. Here the circle always rolls along a straight line.
Can I draw several pens at once?
Yes. Type extra d values, separated by commas, and each pen gets its own curve and colour on the same rolling circle. The preset with five pens shows curtate, cycloid and prolate curves together.
Can I use it on a pen plotter, laser or cutter?
Yes. The SVG download is the drawing at the real size you typed: each curve is one unbroken path, so a plotter draws it 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 and every number is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.