Astroid
Draw an astroid, the four-pointed star curve x = a cos³t, y = a sin³t, three ways that all give the same curve: a small circle a quarter of the size rolling inside a circle, a ladder of length a sliding with its ends on two walls, and the curve’s own tangent lines. Start from one of 13 presets, watch the circle roll, and download a PDF at real size on US Letter or A4, a 300 DPI PNG or an SVG for a pen plotter or laser. 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.
Astroid
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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 astroid formulas by our own code. You may use what you make here, including commercially, with no credit needed.
How to draw and print an astroid
- Pick a preset, or tick what to draw: the curve, the rolling circle, the sliding ladder or the tangent lines.
- Choose the rolling circle (a/4, or 3a/4 rolled the other way: the same curve) and how many ladder positions or tangent lines to draw.
- Tick One quarter only for the classic ladder sliding down a wall.
- 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: parametric curves, envelopes and the sliding ladder problem
- Calculus classes: area and arc length you can check on paper
- Pen plotter, laser engraving and vinyl art
- Posters, cards and wall art
- Star and sparkle shapes with exact curves
- Workshop lessons: the trammel and its moving bar
The 13 presets
| Preset | What is drawn | Turn |
|---|---|---|
| Rolling circle a/4 inside a circle of radius a | curve, rolling circle a/4 | cusps on the axes |
| The curve on its own | curve | cusps on the axes |
| Curve turned 45 degrees | curve | turned 45° |
| Rolling circle 3a/4: the same curve, rolled the other way | curve, rolling circle 3a/4 | cusps on the axes |
| Sliding ladder: 12 positions, with the curve | curve, 12 ladder positions | cusps on the axes |
| Sliding ladder: 24 positions | 24 ladder positions | cusps on the axes |
| Sliding ladder: 48 positions | 48 ladder positions | cusps on the axes |
| Ladder against a wall: one quarter, 16 positions | curve, 15 ladder positions (one quarter) | cusps on the axes |
| Sliding ladder: 90 positions, with the curve | curve, 90 ladder positions | cusps on the axes |
| Ladder against a wall: one quarter, 40 positions | 39 ladder positions (one quarter) | cusps on the axes |
| Tangent lines: 36, cut at the circle | 36 tangent lines | cusps on the axes |
| Tangent lines: 90, with the curve | curve, 88 tangent lines | cusps on the axes |
| Rolling circle, ladder and tangent lines on one drawing | curve, rolling circle a/4, 24 ladder positions, 48 tangent lines | cusps on the axes |
The ladder presets slide a segment of length a with its ends on the two axes. The tangent-line presets draw the tangent at evenly spaced points of the curve, cut where it meets the circle of radius a.
Three ways to draw it
| Construction | How it works | Where it touches the astroid |
|---|---|---|
| Rolling circle | A circle of radius a/4 rolls inside a fixed circle of radius a; a pen on its rim draws the curve | The pen point itself |
| Sliding ladder | A segment of length a with one end on each axis, at angle φ | At (a cos³φ, a sin³φ), a fraction sin²φ of the way along |
| Tangent lines | The tangent at each point of the curve, cut where it meets the circle of radius a | At the point it is drawn from |
All three are drawn in the same frame, so you can lay one over another (the Rolling circle, ladder and tangent lines preset). Each tangent line lies on the same line as the ladder at that angle, which our tests also check.
Size, length and area
| a (fixed circle radius) | Length 6a | Area 3πa²/8 | Rolling circle a/4 | Curve width × height |
|---|---|---|---|---|
| 25.4 mm | 152.4 mm | 760 mm² | 6.35 mm | 50.8 × 50.8 mm |
| 50 mm | 300.0 mm | 2945 mm² | 12.50 mm | 100.0 × 100.0 mm |
| 76.2 mm | 457.2 mm | 6841 mm² | 19.05 mm | 152.4 × 152.4 mm |
The PDF prints a, the length and the area for the size you choose, and a 1 in (or 2 cm) bar to check the print scale.
What prints
| Download | What is in it |
|---|---|
| The page at real size, vector: the drawing, the numbers 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 ladder or tangent line as its own element |
For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| The astroid by rolling circle, ladder or tangents | Astroid (this page) |
| Inside-rolling curves with any number of cusps | Hypocycloid |
| Outside-rolling curves (nephroid and others) | Epicycloid |
| Gear-and-wheel curves with a pen hole inside the wheel | Spirograph Generator |
| A photo turned into thread lines on nails | String Art Generator |
Different tools: the Spirograph Generator draws gear curves with the pen inside the wheel (rounded loops, no cusps), the String Art Generator plans a thread path from a photo, and the Guilloche Pattern Generator draws banknote-style line patterns.
More free tools
Curves, patterns and generative art?
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Frequently asked questions
What is an astroid?
A star-shaped curve with four sharp points (cusps). It is the path of a point on the rim of a circle rolling inside another circle four times as big. Its equation is x = a cos³t, y = a sin³t, or |x|^(2/3) + |y|^(2/3) = a^(2/3), where a is the radius of the big circle.
Where does the name come from?
From the Greek word for star. The name was first proposed, as Astrois, by Joseph Johann von Littrow in 1838.
How does a sliding ladder draw an astroid?
Lean a ladder of length a against a wall and let it slide down. Every position of the ladder touches one curve, and that curve is a quarter of an astroid. With the ladder turned through all four quarters you get the whole astroid. The tool checks every ladder position: it touches the curve at (a cos³φ, a sin³φ), never crosses it, and is exactly a long.
Are the three drawings really the same curve?
Yes. The rolling circle of radius a/4 inside a circle of radius a, the ladder of length a and the tangent lines are drawn in the same frame, and our tests check every point and touch point against x = a cos³t, y = a sin³t to one billionth.
Can a bigger rolling circle draw it?
Yes. A circle of radius 3a/4 rolling inside the circle of radius a draws the same astroid, going round the other way and taking three trips round to close. The tool draws it and our tests check it is the same curve.
What are the area and the length of an astroid?
The area is 3πa²/8 and the whole curve is 6a long, where a is the radius of the fixed circle (the distance from the centre to a cusp). The tool shows both for your size, and our tests check them against a numerical integration.
Is the string-art star made of straight lines from one axis to the other an astroid?
No. Joining point i on one axis to point n − i on the other gives a curve of the form √x + √y = √n, which is part of a parabola. In the astroid’s sliding ladder every line has the same length a; in the string-art star the lines have different lengths. Our tests check that the string-art touch points do not lie on any astroid.
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.