Lissajous Curve
Draw Lissajous curves, also called Lissajous figures or Bowditch curves: the path x = A sin(a t + δ), y = B sin(b t) traced by two waves at right angles. Set the two frequencies, the phase and the height, start from one of 15 presets, view it as an oscilloscope trace and watch it draw. Download the curve as one continuous line in an SVG at real size for a pen plotter or cutter, a 300 DPI PNG, or a PDF on US Letter or A4. 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 is the curve as one unbroken line at the size you typed, ready for a pen plotter, laser or vinyl cutter.
Curve
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Runs in your browser with our own code: nothing you type is uploaded or saved. The curves are drawn from the formula by our own code. You may use what you make here, including commercially, with no credit needed.
How to make and print a Lissajous curve
- Pick a preset or type the x frequency a and the y frequency b (whole numbers 1 to 20).
- Set the phase δ in degrees and the height as a percentage of the width.
- Pick the look (line on white or oscilloscope), the colour and the line width.
- Set the size in inches or cm and the paper, or put all presets on one page.
- Download the PDF and print at 100% (Actual size), or the one-line SVG for a plotter.
What you can use it for
- Physics and electronics lessons: reading frequency ratios on an oscilloscope
- Maths class: parametric curves and ratios
- Pen plotter and laser engraving art
- Logos, posters and wall art
- Embroidery and string art outlines
- Music and sound visuals: intervals as shapes
The 15 presets
| Preset | Ratio a:b | Phase δ | Curve | Touches side / top | Self-crossings |
|---|---|---|---|---|---|
| Circle (1:1, 90°) | 1:1 | 90° | closed | 1 / 1 | 0 |
| Tilted ellipse (1:1, 45°) | 1:1 | 45° | closed | 1 / 1 | 0 |
| Diagonal line (1:1, 0°) | 1:1 | 0° | open (drawn once) | – | – |
| Figure eight (1:2, 0°) | 1:2 | 0° | closed | 1 / 2 | 1 |
| Parabola (2:1, 90°) | 2:1 | 90° | open (drawn once) | – | – |
| 3:2 at 90° | 3:2 | 90° | closed | 3 / 2 | 7 |
| 3:1 at 90° | 3:1 | 90° | closed | 3 / 1 | 2 |
| 3:4 at 45° | 3:4 | 45° | closed | 3 / 4 | 17 |
| 5:4 at 90° | 5:4 | 90° | closed | 5 / 4 | 31 |
| 5:6 at 30° | 5:6 | 30° | closed | 5 / 6 | 49 |
| 7:6 at 90° | 7:6 | 90° | closed | 7 / 6 | 71 |
| 4:5 at 45°, wide | 4:5 | 45° | closed | 4 / 5 | 31 |
| 9:8 at 90° | 9:8 | 90° | closed | 9 / 8 | 127 |
| 13:12 at 60° (dense mesh) | 13:12 | 60° | closed | 13 / 12 | 287 |
| 5:2 at 18° | 5:2 | 18° | closed | 5 / 2 | 13 |
Named presets are exactly that shape, and our tests check it: every point of the circle at the same distance from the centre, the line on y = x, the parabola on x = 1 − 2y², the figure eight crossing itself once at the centre.
Reading the ratio
| Ratio a:b | Touches on each side edge | Touches on top and bottom | Self-crossings (2ab − a − b) |
|---|---|---|---|
| 1:1 | 1 | 1 | 0 |
| 1:2 | 1 | 2 | 1 |
| 1:3 | 1 | 3 | 2 |
| 2:3 | 2 | 3 | 7 |
| 3:4 | 3 | 4 | 17 |
| 3:5 | 3 | 5 | 22 |
| 4:5 | 4 | 5 | 31 |
| 5:6 | 5 | 6 | 49 |
These counts hold for closed curves. Ratios are reduced first, so 2:4 gives the same curve as 1:2.
The formula
| Part | Meaning |
|---|---|
| x = A sin(a t + δ) | Side-to-side wave: A = half the width, a = its frequency, δ = the phase |
| y = B sin(b t) | Up-and-down wave: B = half the height, b = its frequency |
| t from 0 to 2π | One full cycle for a closed curve (after reducing a:b) |
| Points | 160 × (a + b) points, at least 720, drawn as one line |
What prints
| Download | What is in it |
|---|---|
| The page at real size, vector: the curve, the oscilloscope grid if chosen, and a size bar | |
| PNG 300 DPI | The same page as a picture |
| SVG (one line) | Only the curve, one unbroken path at the size you typed, for a plotter, laser or vinyl cutter |
Every page has a 1 in (or 2 cm) bar at the bottom to check the print is at 100%. For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| Two sine waves at right angles (Lissajous figures) | Lissajous Curve (this page) |
| Curves from a wheel rolling in a ring | Spirograph Generator |
| Damped pendulum drawings that spiral inwards | Harmonograph |
| Fine interlaced banknote-style lines | Guilloche Pattern Generator |
| A photo turned into thread lines or one spiral | String Art Generator, Spiral Art Generator |
Different curves: the Spirograph Generator draws rolling-circle curves (hypotrochoids), the Guilloche Pattern Generator draws banknote-style patterns, and the Spiral Art Generator turns a photo into one spiral line.
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Frequently asked questions
What is a Lissajous curve?
The path of a point that moves back and forth in x and in y at the same time, each as a sine wave: x = A sin(a t + δ), y = B sin(b t). The ratio a:b sets the shape and the phase δ tilts and twists it. It is also called a Lissajous figure or a Bowditch curve.
Who discovered Lissajous figures?
The American mathematician Nathaniel Bowditch studied them in 1815. The French physicist Jules Antoine Lissajous studied them in more detail in 1857, using mirrors fixed to vibrating tuning forks, and they carry his name.
How do I read the ratio from the curve?
Count where the curve touches the edges of its box. On a closed curve, the number of touches on one side edge is a and the number on the top edge is b, so the ratio of the x frequency to the y frequency is a:b. This is how an oscilloscope compares two signals. The tool shows both counts, and our tests check them on every preset.
When does the curve close?
Always, here: a and b are whole numbers, so the curve repeats after one full turn of t once the ratio is reduced (6:4 draws the same curve as 3:2). Our tests check that it comes back to its start within one billionth and not sooner.
What does the phase do?
It shifts the x wave against the y wave. At 1:1, 0° gives a diagonal line, 90° a circle (with equal width and height) and anything between a tilted ellipse.
Why do some curves look open, like a line or a parabola?
At some phases the curve runs back over itself, so it looks like a line with two ends. The tool spots these and draws them once, end to end, instead of twice. Open presets: Diagonal line (1:1, 0°), Parabola (2:1, 90°).
How many times does it cross itself?
A closed curve with the ratio a:b (reduced) crosses itself 2ab − a − b times, unless the phase makes it open. A figure eight (1:2) crosses once, 3:2 seven times. The table below lists more, and our tests count the crossings on the drawn path.
Can I draw it on a pen plotter?
Yes. The SVG download is the curve as one unbroken path (one start point, then straight segments), at the real size you typed, so the pen goes down once and lifts once. For a cutter or CNC 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. The curve is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.