Lemniscate
Draw the lemniscate of Bernoulli, the figure-eight curve (x² + y²)² = a²(x² − y²), at any width, with its two foci and the distances from a point to each focus. Switch to the taller lemniscate of Gerono, or to a Cassini oval and watch the curve split into two loops, pinch into a figure eight or round out into an oval as the ratio b/c changes. The tool gives the width, height, area and length of the line, and prints the curve at real size (tiled over sheets if it is bigger than the paper) as a PDF, a 300 DPI PNG or an SVG with each loop as one path. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints across as many sheets as it needs: cut each sheet on its dashed line and butt the edges together, matching the corner marks. The SVG holds the whole drawing at real size.
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Runs in your browser with our own code: nothing you type is uploaded or saved. Every line and number is worked out from the curve equations by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw a lemniscate and print it at real size
- Pick a preset, or choose the curve (Bernoulli, Gerono or Cassini oval) and type the width.
- For a Cassini oval, set the ratio b/c: below 1 gives two loops, 1 the lemniscate, above 1 one loop.
- Tick the foci, the axes, the lines from a point P to both foci, or the Cassini family.
- Pick the line colour and width, the unit and the paper; choose Fitted to one page or Real size.
- Download the PDF and print at 100% (Actual size), or the SVG for a plotter, laser or cutter.
What you can use it for
- Maths lessons: foci, polar curves and the product-of-distances definition
- A figure-eight template at an exact size for crafts, signs and embroidery
- Wall stencils and large outlines on tiled sheets
- Pen plotter, laser and vinyl cutting of exact curves
- Comparing the lemniscate with Cassini ovals and the lemniscate of Gerono
- Line art and logos built on a true mathematical curve
The 14 presets
| Preset | Curve | Width | Height | Area | Length of the line |
|---|---|---|---|---|---|
| Lemniscate of Bernoulli: 100 mm wide, foci and axes | Bernoulli | 100 mm | 35.4 mm | 2500 mm² | 262.2 mm |
| Lemniscate filling the page (200 mm wide) | Bernoulli | 200 mm | 70.7 mm | 10000 mm² | 524.4 mm |
| Figure-eight template: 6 in wide (real size) | Bernoulli | 6 in | 2.12 in | 9 in² | 15.73 in |
| Bold ribbon line: 150 mm wide, 4 mm line | Bernoulli | 150 mm | 53 mm | 5625 mm² | 393.3 mm |
| Wall stencil: 600 mm wide (tiled real size) | Bernoulli | 600 mm | 212.1 mm | 90000 mm² | 1573.2 mm |
| Focal distances: |PF1| × |PF2| at a point | Bernoulli | 120 mm | 42.4 mm | 3600 mm² | 314.6 mm |
| Small outline: 50 mm wide (real size) | Bernoulli | 50 mm | 17.7 mm | 625 mm² | 131.1 mm |
| Lemniscate 20 cm wide with foci | Bernoulli | 20 cm | 7.07 cm | 100 cm² | 52.44 cm |
| Lemniscate of Gerono: 100 mm wide | Gerono | 100 mm | 50 mm | 3333.3 mm² | 304.9 mm |
| Lemniscate of Gerono: 7 in wide (real size) | Gerono | 7 in | 3.5 in | 16.333 in² | 21.34 in |
| Cassini oval, two loops (b/c = 0.9) | Cassini (b/c 0.9) | 120 mm | 36.1 mm | 2284 mm² | 240.3 mm |
| Cassini oval with a waist (b/c = 1.1) | Cassini (b/c 1.1) | 120 mm | 48.8 mm | 4934.3 mm² | 296.4 mm |
| Convex Cassini oval (b/c = 1.5) | Cassini (b/c 1.5) | 120 mm | 74.4 mm | 7427.5 mm² | 317.9 mm |
| Cassini family around the lemniscate (same foci) | Cassini (b/c 1) | 120 mm | 42.4 mm | 3600 mm² | 314.6 mm |
Every value is worked out by the tool from the width. The area counts both loops; the length is the whole line.
Lemniscate of Bernoulli sizes
| Width (2a) | Height (a/√2) | Focus from the centre (a/√2) | |PF1| × |PF2| (a²/2) | Area (a²) | Length (2ϖa) |
|---|---|---|---|---|---|
| 50 mm | 17.7 mm | 17.7 mm | 312.5 mm² | 625 mm² | 131.1 mm |
| 100 mm | 35.4 mm | 35.4 mm | 1250 mm² | 2500 mm² | 262.2 mm |
| 152.4 mm (6 in) | 53.9 mm | 53.9 mm | 2903.2 mm² | 5806.4 mm² | 399.6 mm |
| 200 mm | 70.7 mm | 70.7 mm | 5000 mm² | 10000 mm² | 524.4 mm |
Cassini ovals with foci 50 mm apart
| b/c | Shape | Width | Height | Area | Length |
|---|---|---|---|---|---|
| 0.5 | Two separate loops | 55.9 mm | 6.2 mm | 61.85 mm² | 39.4 mm |
| 0.8 | Two separate loops | 64 mm | 16 mm | 426.79 mm² | 103.6 mm |
| 0.9 | Two separate loops | 67.3 mm | 20.2 mm | 717.73 mm² | 134.7 mm |
| 1 | Figure eight (lemniscate of Bernoulli) | 70.7 mm | 25 mm | 1250 mm² | 185.4 mm |
| 1.1 | One loop with a waist | 74.3 mm | 30.2 mm | 1893.2 mm² | 183.6 mm |
| 1.25 | One loop with a waist | 80 mm | 39.1 mm | 2724.3 mm² | 202.4 mm |
| 1.414 (√2) | One convex loop | 86.6 mm | 50 mm | 3668.7 mm² | 226 mm |
| 1.7 | One convex loop | 98.6 mm | 68.7 mm | 5500.6 mm² | 269.1 mm |
| 2 | One convex loop | 111.8 mm | 86.6 mm | 7729.8 mm² | 315.4 mm |
The foci stay 50 mm apart (c = 25 mm) and only the product b² changes. Areas and lengths of Cassini ovals are worked out numerically by the tool.
The three curves
| Curve | Equation | Shape |
|---|---|---|
| Lemniscate of Bernoulli | (x² + y²)² = a²(x² − y²) | Figure eight, height a/√2 for width 2a, foci at ±a/√2 |
| Lemniscate of Gerono | x⁴ = a²(x² − y²) | Figure eight, height a for width 2a, no foci |
| Cassini oval | |PF1| × |PF2| = b², foci at ±c | Two loops, a figure eight or one loop, by b/c |
What prints
| Download | What is in it |
|---|---|
| The curve fitted to one page or at real size (tiled if needed), with the numbers and a check bar | |
| PNG 300 DPI | The same sheets as pictures (a ZIP when there is more than one) |
| SVG (real size) | Only the drawing: each loop as one closed path, the foci, axes and family as separate parts |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| A lemniscate, Cassini oval or Gerono curve at an exact size | Lemniscate (this page) |
| The ∞ character to copy or type | Infinity Symbol |
| Figures from two waves at right angles | Lissajous Curve |
| Gear-and-wheel loop curves | Spirograph Generator |
| A shape of constant width | Reuleaux Triangle |
Different tools: the Infinity Symbol page copies and types the ∞ character, the Spirograph Generator draws gear curves and the Graph Paper maker prints grids at real size.
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Frequently asked questions
What is a lemniscate?
A figure-eight curve. The best known is the lemniscate of Bernoulli: the points P whose distances to two fixed points F1 and F2 multiply to a constant, chosen so that the curve passes through the midpoint between them. The name comes from the Latin lemniscatus, decorated with hanging ribbons.
What is the equation of the lemniscate of Bernoulli?
With the curve centred on the origin and a half its width: (x² + y²)² = a²(x² − y²). In polar form r² = a² cos 2θ, and as a parametric curve x = a cos t / (1 + sin² t), y = a sin t cos t / (1 + sin² t). Our tests check that all three give the same points to within one billionth of the size.
Where are the foci?
At (±a/√2, 0), so for a curve 100 mm wide they sit 35.36 mm either side of the centre. For every point on the curve the two distances to the foci multiply to a²/2; tick the point P to see the two distances printed.
How big is it?
A lemniscate of Bernoulli of width 2a is a/√2 tall, encloses an area of a² (both loops together) and its line is 2ϖa long, where ϖ = 2.62206 is the lemniscate constant. For a width of 100 mm that is 35.36 mm tall and 262.21 mm long. Our tests check the area and the length against numerical integration to one millionth.
What is a Cassini oval?
The curve of points whose distances to two foci multiply to a constant b², with the foci 2c apart. When b is smaller than c it is two separate loops, when b = c it is the lemniscate of Bernoulli, and when b is larger than c it is one loop: pinched at the waist until b/c reaches √2, convex from there on. Our tests check each of these by b/c. The curves are named after the astronomer Giovanni Domenico Cassini.
What is the lemniscate of Gerono?
A different figure eight, x⁴ = a²(x² − y²), drawn as x = a cos t, y = a sin t cos t. For the same width it is taller than Bernoulli’s (height a against a/√2) and its loops are rounder. It is not a Cassini oval and has no foci, so the foci options switch off for it. Its area is 4a²/3.
Who first studied the lemniscate?
Jakob Bernoulli, in 1694, who described it as a variation on the ellipse: the ellipse uses the sum of the distances to two foci, the lemniscate their product.
How smooth is the line?
Each loop is one path of short straight pieces. The tool keeps splitting them until every piece is within 0.01 mm of the true curve at the printed size, and our tests check that against an independent dense trace of the curve.
How does the real-size print work when the drawing is bigger than the paper?
Choose Real size. The tool splits the drawing over as many sheets as it needs, each with a dashed edge, corner marks and a label such as A1 (row A, column 1). Cut each sheet on its dashed line, butt the edges together in order and tape them. Every printed point is at its true position to within 0.2 mm in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar on each sheet lets you confirm the scale.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every line and number is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.